The "what if"-factor is one of the most inspiring things for fanfiction. How often do we think "what if" after reading a book or watching a movie or an episode of a tv series? And it's one of the things that brings people to table-top roleplaying - in a campaign of Game of Thrones or Star Wars, you can explore the world and the people in it.
But fanfiction is still far from being mainstream. Tell people you write short stories or poetry - totally accepted by most people. But tell them about fanfiction and the muggles look at you like an exhibit at the cabinet of curiosities ;)
And reading fanfiction improved my English. Reading stories and wondering "is that correct?", finding all the "their" and "there" and "they're" in a wild mix, all that helped me in simply writing: even if I don't find the best term for things, people get what I want to say.