This section, the last section in chapter 1, has 4 verses.

To set the stage, let us read the previous four verses.

Titus 1:9-12
9 Holding fast the faithful word as he hath been taught, that he may be able by sound doctrine both to exhort and to convince the gainsayers.
10 For there are many unruly and vain talkers and deceivers, specially they of the circumcision:
11 Whose mouths must be stopped, who subvert whole houses, teaching things which they ought not, for filthy lucre's sake.
12 One of themselves, even a prophet of their own, said, The Cretians are alway liars, evil beasts, slow bellies.

Now to the last four verses of the chapter

Titus 1:13-16
13 This witness is true. Wherefore rebuke them sharply, that they may be sound in the faith;
14 Not giving heed to Jewish fables, and commandments of men, that turn from the truth.
15 Unto the pure all things are pure: but unto them that are defiled and unbelieving is nothing pure; but even their mind and conscience is defiled.
16 They profess that they know God; but in works they deny him, being abominable, and disobedient, and unto every good work reprobate.

We will begin with the Barclay. This commentary covering all four verses [13-16].  The first three sentences are verses 13-16 paraphrased.

THE PURE IN HEART
For that very reason correct them with severity, that they may grow healthy in the faith and not pay attention to Jewish fables and to rules and regulations made by men who persist in turning their backs on the truth.

‘To the pure, all things are pure.’

But, to those who are defiled and who do not believe, nothing is pure, because their mind and conscience are defiled. They profess to know God, but they deny their profession by their deeds, because they are repulsive and disobedient and useless for any good work.

THE great characteristic of the Jewish faith was its thousands of rules and regulations. Many things were branded as unclean; many kinds of food were classified as forbidden. When Judaism and Gnosticism joined hands, even the body became unclean, and the natural instincts of the body were held to be evil. The inevitable result was that long lists of sins were constantly being created. It became a sin to touch certain things; it became a sin to eat certain foods; it even became a sin to marry and to have children. Things which were either good in themselves or quite natural were considered unclean.

It is the heart which makes all the difference. If someone is pure in heart, all things are pure to that person. If someone is unclean in heart, then everything that person thinks about or speaks about or touches becomes unclean. This was a principle which the great classical writers had often stated. ‘Unless the vessel is pure,’ said Horace, ‘everything you pour into it grows bitter.’ Seneca said: ‘Just as a diseased stomach alters the food which it receives, so the darkened mind turns everything you commit to it to its own burden and ruin. Nothing can come to evil men which is of any good to them, nay nothing can come to them which does not actually harm them. They change whatever touches them into their own nature. And even things which would be of profit to others become pernicious [harmful] to them.’ The person with a dirty mind makes everything dirty, taking even the loveliest things and covering them with smut. But the person whose mind is pure finds purity in everything.

It is said of these people that both their minds and their consciences are contaminated. We come to decisions and form conclusions by using two faculties. We use intellect to think things out; we use conscience to listen to the voice of God. But if the intellect is warped in such a way that it sees the unclean everywhere and in all things, and if the conscience is darkened and numbed by continual consent to what is evil, it becomes impossible to take any good decision at all.

If we allow impurity to infect the mind, we will see all things through a mist of uncleanness. The infected mind soils every thought that enters into it; the imagination turns every picture which it forms into a source of lust. Every motive is misinterpreted. Every statement is given a double meaning. To escape that uncleanness, we must walk in the cleansing presence of Jesus Christ.

THE UGLY AND THE USELESS LIFE
WHEN people get into this state of impurity, they may know God intellectually but their lives are a denial of that knowledge. Three things are singled out here about such people.

(1) They are repulsive. The word (bdeluktos) is the word particularly used of idols and images. It is the word from which the noun bdelugma, an abomination, comes. There is something repulsive about someone with an obscene mind, who makes sniggering jokes and whose comments are full of innuendo.

(2) They are disobedient. Such people cannot obey the will of God. Their consciences are darkened. They have made it almost impossible for themselves to hear the voice of God, let alone obey it. People like that cannot be anything but an evil influence and are therefore unfit to be instruments in the hand of God.

(3) That is just another way of saying that they have become useless to God and to others. The word used for useless (adokimos) is interesting. It is used to describe a counterfeit coin which is below standard weight. It is used to describe a cowardly soldier who fails in the testing hour of battle. It is used of a candidate rejected for public office, someone whom the citizens regarded as useless. It is used of a stone which the builders rejected. (If a stone had a flaw in it, it was marked with a capital A, for adokimos, and left to one side, as being unfit to have any place in the building.) The ultimate test of life is usefulness, and those who are a bad influence and who have a contaminating effect are of no use to God or to other people. Instead of helping God’s work in the world, they hinder it; and uselessness always invites disaster. ~Barclay Commentary

Now to the other commentaries, beginning with the general and going to the specific.

We will begin with the Matthew Henry Main where it is speaking to verse 13. I am extracting this from a commentary covering verses 6-16

3. Here is the verification of this by the apostle himself: Titus 1:13. This witness is true, The apostle saw too much ground for that character. The temper of some nations is more inclined to some vices than others. The Cretans were too generally such as here described, slothful and ill-natured, false and perfidious, as the apostle himself vouches. And thence,

4. He instructs Titus how to deal with them: Wherefore rebuke them sharply. When Paul wrote to Timothy he bade him instruct with meekness; but now, when he writes to Titus, he bids him rebuke them sharply. The reason of the difference may be taken from the different temper of Timothy and Titus; the former might have more keenness in his disposition, and be apt to be warm in reproving, whom therefore he bids to rebuke with meekness; and the latter might be one of more mildness, therefore he quickens him, and bids him rebuke sharply. Or rather it was from the difference of the case and people: Timothy had a more polite people to deal with, and therefore he must rebuke them with meekness; and Titus had to do with those who were more rough and uncultivated, and therefore he must rebuke them sharply; their corruptions were many and gross, and committed without shame or modesty, and therefore should be dealt with accordingly. There must in reproving be a distinguishing between sins and sins; some are more gross and heinous in their nature, or in the manner of their commission, with openness and boldness, to the greater dishonour of God and danger and hurt to men: and between sinners and sinners; some are of a more tender and tractable temper, apter to be wrought on by gentleness, and to be sunk and discouraged by too much roughness and severity; others are more hardy and stubborn, and need more cutting language to beget in them remorse and shame. Wisdom therefore is requisite to temper and manage reproofs aright, as may be most likely to do good. Jude1:22-23, Of some have compassion, making a difference; and others save with fear, pulling them out of the fire. The Cretans' sins and corruptions were many, great, and habitual; therefore they must be rebuked sharply. But that such direction might not be misconstrued. ~Matthew Henry Main

Quoted verses:
Jude 1:22-23 ...verses speaking to members caught up in apostasy
22 And of some have compassion, making a difference:
23 And others save with fear, pulling them out of the fire; hating even the garment spotted by the flesh.

Now to the Matthew Henry Concise which covers verses 10-16,

False teachers are described. Faithful ministers must oppose such in good time, that their folly being made manifest, they may go no further They had a base end in what they did; serving a worldly interest under pretence of religion: for the love of money is the root of all evil. Such should be resisted, and put to shame, by sound doctrine from the Scriptures. Shameful actions, the reproach of heathens, should be far from Christians; falsehood and lying, envious craft and cruelty, brutal and sensual practices, and idleness and sloth, are sins condemned even by the light of nature. But Christian meekness is as far from cowardly passing over sin and error, as from anger and impatience. And though there may be national differences of character, yet the heart of man in every age and place is deceitful and desperately wicked. But the sharpest reproofs must aim at the good of the reproved; and soundness in the faith is most desirable and necessary. To those who are defiled and unbelieving, nothing is pure; they abuse, and turn things lawful and good into sin. Many profess to know God, yet in their lives deny and reject him. See the miserable state of hypocrites, such as have a form of godliness, but are without the power; yet let us not be so ready to fix this charge on others, as careful that it does not apply to ourselves. ~Matthew Henry Concise

Now a couple of things from the Biblical Illustrator.

Rebuke them sharply
Here we have another adoption of the phraseology of health or “soundness” in relation to the faith. Probably it was suggested to the apostle by the previous adoption of phrases indicative of disease, and of severe remedies. A sharp knife, instruments of cautery [kaw-tuh-ree] [hot iron used to destroy tissue], firm handling, free incisions, are needed for some poisonous and putrefying sores; and as in former days Titus had to show the Corinthians how to purge out the old leaven, to deliver wicked persons to Satan, to rebuke pretentious sciolism [sahy-uh-liz-uh m] [superficial knowledge] and proclaim “no quarter” to certain kinds of vice, so once more he had to lift up his voice like a trumpet, and out of sheer kindness was commanded not to spare them. ~Biblical Illustrator

Different modes of dealing with different sins
According to the nature of sins and sinners we must set an edge upon our reproofs and sharpen them; for all sins are not of one size, nor all sinners of one strain; but some sins are more enormous than others, and some sinners are more obstinate than others. Some sins are of ignorance, some of malice; some secret, some open; some sinners are as wax to work on; some are stony and stiff-necked; some have here and there their freckles and frailties on them: others are spotted all over like leopards, or, like the Ethiopian, they never change their hue; no washing doeth them good. Now, we must wisely put a difference between both. Compassion must be showed upon some; and others, whom love cannot allure, fear must force. Some must be saved by love, and some be pulled out of the fire. Some sores need but a gentle lenitive, some a sharper drawer; some require but the prick of a needle to open them, others a more painful lancing and cutting; and some a cutting off. ~Biblical Illustrator

Christian reproof
I. Christian reproof should always be based on a certain convicting. Mere hearsay insufficient; general rumor unreliable. Inquisitorial curiosity different from faithful watchfulness.

II. Christian reproof should be thorough and effective. A cutting rebuke need not be unkind. Sarcasm, satire, scorn—these are unbecoming a Christian teacher. Soft words break hard hearts; warmth melts, while coldness freezes.

III. Christian reproof should be for the sinner’s good—“That they may be sound in the faith.”
Wrong motives:
1. To save appearances.
2. To maintain dignity.
3. To gratify revenge.

Right motives:
1. To save the purity of the Church.
2. To prevent the spread of contagion.
3. To restore to spiritual life and privilege. ~Biblical Illustrator

Sharp rebuke sometimes needed
The words is a metaphor taken from surgeons, who cut out dead flesh to the quick, but it is in order to healing. Cutting words have done great cures: many a diseased, festered soul has been made sound, both in faith and manners, by severe reprehension [reproof]. Learn hence, that although, generally speaking, we ought to temper our reproofs with much gentleness and meekness, yet there is a time when we must reprove sharply, that men may be “sound in the faith.” We may, we must speak cutting words when kind words will not do. ~Biblical Illustrator

Now to the specific commentaries.

This verse is broken out in three parts:

1] This witness is true.
2] Wherefore rebuke them sharply.
3] That they may be sound in the faith.

1] This witness is true.  ...this phrase speaking of the character of the Cretians

This witness is true - That is, this testimony long before borne by one of their own number, was true when the apostle wrote to Titus. The fact that this was the general Character of the people, was a reason why he should be on his guard in introducing men into the ministry, and in the arrangement of affairs pertaining to the church. That it was true. ~Barnes Notes

This witness is true - What Epimenides [ep-ih-MEN-ih-deez] said of them nearly 600 years before continued still to be true. Their original character had undergone no moral change. ~Adam Clarke

This witness is true - The apostle confirms what the poet had said; he knew it to be fact from his own experience, and by the observation he had made when in the island: he does not say, that all that Epimenides [ep-ih-MEN-ih-deez] had said, in the poem referred to, was true; but this character, which he had given of the Cretians, and which he cites, and uses to a good purpose; from whence it may be observed, that the writings of the Heathen poets may be read with profit, and be used to advantage, if carefully and prudently attended to; for what is truth, let it come from whom, or by what means it will, ought to be received. ~John Gill

2] Wherefore rebuke them sharply.

Wherefore rebuke them - Notes, 2 Timothy 4:2.

2 Timothy 4:2 [see Lesson]
Preach the word; be instant in season, out of season; reprove, rebuke, exhort with all longsuffering and doctrine.

Sharply - ἀποτόμως apotomōs - “cuttingly, severely” - from ἀποτέμνω apotemnō, “to cut off.” The word is used here in the sense of severity, meaning that the reproof should be such as would be understood, and would show them plainly the wickedness of such traits of character. He was not to be mealy-mouthed, but he was to call things by their right names, and not to spare their faults. When men know that they are doing wrong, we should tell them so in few words; if they do not know it, it is necessary to teach them, in order to convince them of their error. ~Barnes Notes

Rebuke them sharply - Αποτομως· Cuttingly, severely; show no indulgence to persons guilty of such crimes. ~Adam Clarke

Wherefore rebuke them sharply - not merely upon the testimony of the poet, but upon the confirmation of it by the apostle; and not because of these general and national characters, but because these things personally and particularly belonged to the persons before described; whom the apostle would have rebuked, both for their bad principles, teaching things that they ought not; and for their immoralities, their lying and deceit, their intemperance, luxury [wrongly handled], and idleness, things very unbecoming the Christian name; and therefore since their offences were of an heinous nature, and they lived in them, and were hardened and obstinate, and were like to have a bad influence on others, they must be rebuked "sharply": rebukes ought to be given according to the nature of offences, and the circumstances of them, and the offenders; some are to be given privately, others publicly; some should be reproved with gentleness and meekness, and be used in a tender and compassionate way; others more roughly, though never in a wrathful and passionate manner, yet with some degree of severity, at least with great plainness and faithfulness; laying open the nature of the evils guilty of in all their aggravated circumstances, without sparing them in the least; doing, as surgeons do by wounds, though they take the knife, and use it gently, yet cut deep, to the quick, and go to the bottom of the wound, and lay it open: and so the phrase may be rendered here, "rebuke them cuttingly"; cut them to the quick, and spare them not; deal not with them as Eli with his sons, 1 Samuel 2:23 but speak out, and expose their crimes, severely reprove them, that others may fear. ~John Gill

Quoted verse:
1 Samuel 2:23
And he said unto them, Why do ye such things? for I hear of your evil dealings by all this people.

3] That they may be sound in the faith.

That they may be sound in the faith - That they may not allow the prevailing vices to corrupt their views of religion. ~Barnes Notes

That they may be sound in the faith - That they may receive the incorrupt doctrine, and illustrate it by a holy and useful life. ~Adam Clarke

That they may be sound in the faith - that they may be recovered from their errors, to the acknowledgment of the truth; that they may receive the sound doctrine of faith, the wholesome words of Christ, and speak the things which become them, and use sound speech, which cannot be condemned; and that they may be turned from their evil practices, and appear to be sound, as in the doctrine, so in the grace of faith; or that that by their works may appear to be genuine, true, and unfeigned; and that they may be strong and robust, hale and healthful, and not weak and sickly in the profession of their faith. Rebukes being to persons infected with bad principles and practices. ~John Gill

So, the church sees immorality, lying or false doctrine being held or preached and action is taken.  In such cases, strong correction and rebuke is needed.  The correction and rebuke is applied to bring the member back to strong faith and obedience to God; to get the brother/sister back to moving forward in the salvation process.  This is the lesson of verse 13.