Ideas and stories hit me and I feel the need to share them. When I started writing Mario fanfiction, the main driver behind my work was to try and share my ideas about character backgrounds and other unexplained things about the series (almost self-serving in a way), but that morphed into wanting to share stories for the sake of the stories (a purer motivation than the initial one). And, once stories have started, not wanting to feel like a quitter or let the readers down by giving up and never finishing is a huge motivator, but not really an inspiration.
Of course, there's the fact that I enjoy the writing process too. My best writing experience hands-down was a fanfic idea randomly popping into my head one day and refusing to let me concentrate on anything else until I had hammered it out. I published it and want people to read it and love getting reviews and all that, but while I was writing it, I was writing it for me. It was the most fun, most incendiary and most satisfying writing experience I've ever had, and I hope I'll have more like it.