What does Matthew 7:3 mean?

Earlier verses contained a statement which is easily—and often—taken out of context. Jesus has commanded that His followers "judge not" (Matthew 7:1), but immediately begun to explain what this really means (Matthew 7:2). In short, what Christ condemns is shallow, hypocritical, or selfish criticism of others. Christ is clear that "right judgment" (John 7:24) includes distinguishing between good and evil (Matthew 7:15–18). Judging others as if we, ourselves, are the standard of goodness is evil.

God doesn't give human beings authority to judge the righteousness of others using our own preferences as the standard. None of us are perfectly righteous ourselves. We are in no position to casually pronounce judgment because we ourselves can be guilty of sin.

Jesus shows in this verse that we are often blind to our own sins. This makes shallow condemnation towards others even more foolish. This again uses the technique of exaggeration for effect, known as hyperbole. Jesus asks why we are so skillful to see a speck in a brother's eye and so unable to notice the log in our own eye. It seems human beings can recognize the smallest of sinful infractions in the lives of others while walking around with obvious and ugly sins of their own. This is human nature, and it is why God forbids His people to assume His role as judge over their brothers and sisters.

It's crucial to realize, however, what Jesus says in verse 5. Once the "log" has been removed from our eye, we are told to then help remove the speck from our brother's eye. The difference is that in helping our brother in this way, we are no longer attempting to be his judge. We are acknowledging the reality of sin, submitting to it ourselves, and serving our brother by helping him. Likewise, those "specks" and "logs" are truly sinful; the point is not that everything we see in others must be accepted, but that we should approach sin with humility and grace, not arrogance.