A good piece of fanfic fills in the blanks that naturally exist in a show or movie. It expands on the source material/character in a believable way. It makes me think, "Yeah, that IS how it was." Even if I'd never considered it COULD be like that until I read the story.

I've always felt that shows and movies paint in broad strokes, while the written word can go in and paint the fine details and gradations. What I love most about writing and reading fanfic is the chance to delve more deeply into the characters I love.