Pretend for just a minute. Pretend that you discovered the cure for cancer. If that happened, would you (a) keep the information to yourself, or (b) make sure everyone possible knows about it. If you are a decent person, the obvious answer is b.
So apply that logic to the person who gets saved and wants to share the good news with everyone. The evangelizing believers are merely doing what their hearts direct them to do out of concern for everyone.
It vexes me that the unbelieving world has such a problem with believers talking about their beliefs, when the fact is, the believers are only doing what one would expect, if their beliefs are true. It should come completely natural for someone to share such important information with others. The believers who keep the secret to themselves are either frauds, or very cruel people who don't want anyone else to be saved. Again, just imagine someone being given the secret to cure cancer, and deciding not to tell anyone about it for whatever reason.
I'm not saying anything that isn't obvious but sometimes I get the idea that the unbelieving world doesn't think this through. They just assume that anyone who tries to tell them how to save their soul is just doing it because they want to feel superior, or because someone in their church told them to do it.
"But let us not come up with any patronizing nonsense about Jesus being a great teacher. He has not left that open to us. He did not intend to." C.S. Lewis