Why do you look at the speck of sawdust in your brother's eye and pay no attention to the plank in your own eye? How can you say to your brother, 'Let me take the speck out of your eye,' when all the time there is a plank in your own eye?
This is one of those verses that we like to quote to others when they are confronting us with issues in our own lives.
"Don't you tell me! I know you! I know what you do, get the beam out of your own eye!"
Nobody likes to be confronted. It is difficult, and painful at times. But God has designed this life to be lived in fellowship with others, and part of true fellowship is confronting your brothers and sisters when they are in need of it.
These two verses are there to help us make that confrontation process a little less painful. I used to think that this meant that I had to be near-perfect before I could confront anyone, but that isn't true. If it were, there would never be any confrontation because none of us are ever perfect. What I believe this is talking about is that it is much easier to take confrontation from someone who has admittedly messed up and made it right.
Getting the plank out of your eye is to clear yourself of wrong by asking forgiveness of others, and working to make things right. If you are doing this all of the time, you will not look perfect to others, but you will look like someone who is willing to make things right, as humbling as it is.
So many times I want to hide my mistakes, but I have found that as a teacher, I have had the most success when I confess when I have been lazy, or when I have been a slacker. And when I talk about my other struggles, it really opens the students up. They realize that they are not alone. Some of my most amazing opportunities have been when I really screwed things up, and had to admit my wrong-doing.
Getting the plank out of your eye does not mean you have to be perfect, it means you are willing to admit that there is a big board stuck in your eye and that if you can get the plank out, by admitting your wrongs, then they can get the speck out as well. You have already paved the way.