For some weird reason, I really enjoy reading and editing papers. Part of it is that I'm good at it, but I think the other part is that it's a source of pride - that finished work would not be as perfect without your help. (So it's really embarrassing if you miss something and your friend/tutee hands in a paper with mistakes on it. This is why I re-read things after I proofread to make sure I didn't miss anything.)
I think another part of the proofreading process that's fun is learning about what that person is writing about. One of the students I tutor was telling me about this great project she's working on for her social studies class: a series of journal entries, letters, supply requests, and other documents needed to start a colony in the New World and report on its success to Queen Elizabeth I. I'm a huge history nerd so I was really excited to hear about the project and I definitely want to help with proofreading, if only because I am dying to see what she writes about!
But mostly, I just find it soothing to pick through words and fix things. It's a game for me, and my winning streak is pretty high at this point.