I have mixed feelings about fan fiction. I've written some myself. 99% of those will never be seen. So how do I feel about it?
As a Fan:
It's fun: You can think about your favorite characters and put your ideas to paper. It's a homage in a way. But even when I was writing mine, it didn't feel right. It must have been the writer in me. I liked doing it. But the characters I came up with, became more interesting to me.
It's practice: What a great writing exercise. You can try out a lot of different "what if" plots. Some of the better fan fiction communities give you great feedback. I didn't use them. I didn't post much.
It's not yours: Legal issues aside. The setting you love so much, is not yours. You are only borrowing it. As a budding writer it did bother me in the back of my head. I had to stick to someone else's rules for the world I was using.
Sometimes you get inspired: I have at least four characters and one story that came out of the fan fiction I wrote, that are original. That's the reason I stopped writing it. I wanted to write more about Eloise (she came out of an X-man fan fiction) or my disappearing island (Don't want to say which one, but it became horror. When the original setting was not horror). I really think it helped me come up with my own universes.
A nice way to breath life into a dead franchise: Two of the stories I put up on the internet were for the Legacy of Kain series of video games. A wonderful, gory, horror game series. One that ended with a few major questions never addressed. There are still avid fans of the game and writing the stories (some silly, some serious) is a way to keep it going for the fans.
As an Author:
It's proof you "made it": When I discover a piece of fan fiction online based on a story I wrote. To me it is proof my work is loved enough to inspire someone else. No matter how bad it is, I would still feel honored.
It is mine! I will not be like some authors and hunt down anyone that uses their characters and settings. There is one situation in which I feel that I would have to ask a fan fiction writer to stop. Trying to make money off my characters.
I can only think of one instance where that happened. I was in a fan fiction group. And one woman had just bought a book from Amazon, in which the story was very familiar. The characters were as well. A little research showed that the author of the book did not hide the fact it was a fan fiction. In this case the author of the original work did not have to step in and do anything. Other fan fiction writers contacted the author of the fan fiction book and asked her to stop. Most do not want to make money off their fan fiction and really do not like anyone who tries.
A way to pass the flame: Robert Bloch got his start in what would be considered fan fiction now. He loved HP Lovecraft stories. So much that he contacted Lovecraft for permission to publish some of his stories that were set in the Cthulhu mythos wikipedia. Bloch did ask permission and even added to the mythos.
I understand the authors that want to hunt down anyone that uses their characters. I understand the ones that cringe when they hear about a fan fiction story. But for me personally, I won't be upset. Heck I might even encourage it. Think about how many other budding writers there are out there, that just need the encouragement to break off and make their own worlds. If something I wrote can be the seed, I would would be very proud.