How should I remove my reverse switch? (Picture attached)

My car is a 5sp manual 2009 aveo

Attachment 11896

The reverse switch has looked like this since I got the car. Just today it started giving me problems. It only works when someone is pushing it in all of the way.

How should I go about removing it, since the socket is destroyed? I don't want to start until I have a good idea of how to go about this. I don't want to mess it up even more to where I can't remove it at all.

Also, how does this even happen? Is it possible that battery acid did this? I'm interested to see what would have caused this to happen. I'm also curious as to what you even call this. Would it just be called a split open socket, or is there a name for something like this?