Stephen King (who often writes about the writer's muse, even in his fiction) once wrote that all authors process their stories differently. Picture a horse standing next to a pond, and how two different authors might write a story about that scene: Louis L'Amour would write something about the horse's owner stopping to get a drink before heading out into the open prairie. Stephen King would write about a creature coming out of the pond and eating the horse.
Neither is right or wrong; they just process it differently.
I think it's the same for fanfic authors. I've written some myself. You enjoy a story, you like thinking about it, and you want to work out a problem or an issue or a question in fiction, rather than discussing it in a forum like this one. It's just a different way of processing. You like a character, you think, "What would this character think about so-and-so," and you try to get inside that character's head and write about it.
It's not a matter of "use your own characters." It's really not that simple. In some cases, I would say that the authors are simply plugging in the canon characters into a story of their own making, but such fan fictions are usually not very popular, because the characters don't act like the readers expect.