"But I say to you, do not resist an evil person, but whoever slaps you on the right cheek, turn the other to him also"
The idea seems to be that people had the idea that they needed to be the source of vengeance in their own lives. He is saying that you have to get over this vengeance thing. It will eat you alive. You don't have to pay people back for what they do to you. God will handle vengeance.
Each one of the things in this list are things that an employer or government official or private citizen could force you to do. Jesus is saying give them more so that you don't have this big fight on your hand. And most importantly, don't get into the business of trying to pay back the evil they did to you. Instead of payback, find a way to go the second mile. Find a way to give them more than they asked for. They don't deserve the extra, but you are going to give it rather than the eye-for-an-eye payment that you want to give them.
Rebuild the relationship instead of creating a permanent war.