'You have heard that it was said, "An eye for an eye and a tooth for a tooth". But I say to you, do not resist an evildoer. If anyone strikes you on the right cheek, turn to him the other also.'
So in other words even though there are shitty people in the world then anyone who considers themselves a christian should strive to not seek revenge against such people by killing them or wishing them to be killed. Instead they should forgive and leave any judgment for the lord.
Funny thing about this quote. I had an evangelical protestant friend who offered a bit of context for that statement, and it doesn't necessarily mean to forgive. Note that it says "strike you on the right cheek". In order to do that, they will have hit you with their left (non-dominant) hand. So that would be a slap or a weak punch, a bullying-type attack. By "turning the other cheek" you wordlessly challenge them to an even fight. With that interpretation it was more a dare to escalate the situation to a full on brawl, a subtle "come over here and say that".
He also gave me an interesting take on the "walk a mile in another man's shoes" one as well, but that's even less relevant.