Do you see, that He forbids not judging, but commands to cast out first the beam from your eye, and then to set right the doings of the rest of the world? For indeed each one knows his own things better than those of others; and sees the greater rather than the less; and loves himself more than his neighbor. Wherefore, if you do it out of guardian care, I bid you care for yourself first, in whose case the sin is both more certain and greater. But if you neglect yourself, it is quite evident that neither do you judge your brother in care for him, but in hatred, and wishing to expose him. For what if he ought to be judged? It should be by one who commits no such sin, not by you.
Thus, because He had introduced great and high doctrines of self denial, lest any man should say, it is easy so to practise it in words; He willing to signify His entire confidence, and that He was not chargeable with any of the things that had been mentioned, but had duly fulfilled all, spoke this parable. And that, because He too was afterwards to judge, saying, “Woe unto you, Scribes and Pharisees, hypocrites.” Yet was not he chargeable with what has been mentioned; for neither did He pull out a mote, nor had He a beam on His eyes, but being clean from all these, He so corrected the faults of all. “For it is not at all meet,” says He, “to judge others, when one is chargeable with the same things.” And why marvel at His establishing this law, when even the very thief knew it upon the cross, saying to the other thief, “Do you not fear God, seeing we are in the same condemnation;” expressing the same sentiments with Christ?
But you, so far from casting out your own beam, dost not even see it, but another's mote thou not only see, but also judgest, and essayest to cast it out; as if any one seized with a grievous dropsy, or indeed with any other incurable disease, were to neglect this, and find fault with another who was neglecting a slight swelling. And if it be an evil not to see one's own sins, it is a twofold and threefold evil to be even sitting in judgment on others, while men themselves, as if past feeling, are bearing about beams in their own eyes: since no beam is so heavy as sin.
His injunction therefore in these words is as follows, that he who is chargeable with countless evil deeds, should not be a bitter censor of other men's offenses, and especially when these are trifling. He is not overthrowing reproof nor correction, but forbidding men to neglect their own faults, and exult over those of other men.
For indeed this was a cause of men's going unto great vice, bringing in a twofold wickedness. For he, whose practice it had been to slight his own faults, great as they were, and to search bitterly into those of others, being slight and of no account, was spoiling himself two ways: first, by thinking lightly of his own faults; next, by incurring enmities and feuds with all men, and training himself every day to extreme fierceness, and want of feeling for others.
Source: Homilies on the Gospel of St. Matthew (New Advent)