In order to illustrate the problem with judging others. Jesus uses a humorous, even absurd, exaggeration. It makes no sense to condemn someone’s small error if you have a larger error in your own life.
The word usually translated “speck” (κάρφος) is a bit of straw or word, a small splinter, or even “a tiny foreign object in a wine cup” (BDAG). The word translated “log” (δοκός) is a heavy beam used to construct a roof or to bar a door (BDAG). In Josephus’s Jewish War, this word is used to describe a Roman battering ram (JW 3.124).
The contrast is therefore between a tiny insignificant thing, maybe something that is irritating but not really that noticeable and a massive piece of wood that is impossible to miss. John Nolland says this is a scene of “grossly selective perspective” (Nolland, Matthew, 320). The hypocrite only sees one thing, perhaps an issue they consider to be the most important issue of them all, but it is a mere speck compared to a major sin (likely hypocrisy itself) in their own life.
If we deal with “the log in our own eye” are we permitted to condemn others for their speck? Probably not. Jesus is certainly exaggerating, and has been described as ironic or even sarcastic here (Geulich, Sermon on the Mount, 352-3). There are several important observations to be made here.
First, if one is able to actually see what sin they do have in their lives, then they should be more concerned with dealing with their own sin than nit-picking minutia in another person’s life. There is something essentially hypocritical about pointing out another person’s sins when you are unwilling to deal with the same sin you your own life.
Second, if one is dealing with their own sin, they ought to be more sensitive toward people with similar problems. This is of course not always the case, especially if a person is guilty of a “grossly selective perspective.” Jesus’s disciples need to deal with sin, but knowing the extent of one’s own shortcomings must lead to a sensitive and gentle correction.
Third, this is not “blanket tolerance or moral indifference” (McKnight, Sermon on the Mount, 230). Jesus is not saying “nothing offends God so love everyone and everything they do.” He is saying that in an ideal Christian community, there will be enough love and grace among the brothers and sisters that condemning one another will not even be a possibility.
Fourth, the follower of Jesus needs to think about the impact of our condemnation of sin. If a person has been caught in sin and is publically shamed as a result, that is not permission to pile on our own gossip and rage toward the person. Likely as not, they are going through a personal hell as their life falls apart, the individual brother or sister in Christ does not need to fan the flames (probably through verbal sins of their own).
The result taking one’s own problem is that we will “see clearly” (Τότε διαβλέψεις). “See clearly” it to have one’s eyes open wide and looking intently at something (BDAG), perhaps with a clear understanding of what is being seen. Perhaps Jesus is suggestion the one who has dealt with a particular problem can gently correct a fellow disciple since they have experienced the same forgiveness. In fact, the very fact the Lord’s Prayer includes forgiveness ought to be a warning against jumping to judgment too quickly.
Maybe you have had an experience where someone was picking at some minor problem in your life and you knew the person was a hypocrite. Sadly this often is a parent, a pastor or teacher. It may have been a case of “do as I say, not as I do.” My sense is most people will read Jesus’s exhortation to “remove the log from our own eye” as referring to those encounters with hypocrites. But I do not see Jesus narrowing this down to only “those other people.” He tells his closest disciples they need to focus on their sin rather than looking for specks to pick out of someone else’s eye.