There are certain kinds of judgements we need to make each and every day. We must judge if the person or people approaching us on the street or in their cars are likely to be safe or not. Is the salesman being honest? Then there's election season ... but let's leave that one alone. Some judgments help us stay safe and navigate the world we live in successfully.
However, all too often we are encouraged by society to be hypercritical of everyone and make snap judgements about people based on ... almost no facts at all, skewed perspectives based on a variety of things including "how it used to be" (which is often viewed through rose-colored glasses ... working with the whole vision metaphor here ... and therefore never was quite that way), or on gossip among other possibilities. Lots of our entertainment today is based on harsh, critical, angry judgment, the very sort of thing we are being warned against in this passage.
Given this calling from society to play harsh judge, these verses are extremely useful for us today. Jesus calls us not to judge others. We are warned that the way we judge others is the way we will be judged. This is not simply dealing with judgment we face before Jesus at time's end. No, we are warned that in the here and now to judge others harshly will have a serious backlash. Those who judge others the most severely are frequently surprised when they are treated no more generously by those who know them than were the victims of their own harsh assessments. Jesus calls us to steer clear of this trap. It's wonderful advice.
He softens the blow with the wonderful joke about the log in your eye. But there is also a tempering message here. We are to mind our own business, deal with our own flaws, and let others worry about their own specks. It's excellent advice, advice to live by, advice to grow a loving community with if we care to be so countercultural and take it.
Oh, and by the way, "you hypocrite" may well refer to ... according to scholars ... Greek actors for the stage, meaning you are not to act out your faith without really meaning it. An interesting interpretation in this context.