Fanfiction is a love letter.
It's a reader's/player's/participant's expression of appreciation and joy for the imagination of the original author(s) and creators of a fictional world/universe.
It is a writer's way (whether amateur or experienced) of giving something back to that fictional world in the form of the moments we enjoyed the most, the characters we loved, and the stories that made a difference to us.
Fiction is a means of escapism--we sometimes see the world around us as not so challenging or difficult when we can share in the struggles of a character who is facing far greater foes than we are. It gives us someone to pull for, to follow. To care about, and even to hate sometimes. A well-crafted book is an adventure you can put in your bag and open at any time.
Perhaps that is why the game of Myst appeals to me.
Fanfiction is equally escapism, but one that the 'fan' can create in reply to the original fiction. We all love the fictional worlds we do for various reasons, and those reasons are often deeply personal. Something about those stories or characters harmonizes with us, and we want to continue to hold on to that feeling. In rarer circumstances, sometimes we want to express the frustrations we have when stories don't go the way we want. Or if we feel the story ended too soon.
But this is also another expression of love: we want what is best or right for the fictional people involved. It's a writer's response in a more eloquent form than an angry tweet or electronic comment. Plus, it's really a credit to the original creator: they have made a world so important to the reader as to make the reader-turned-author want to fight for a different ending.
So writers, do what you do. And fans, love what you love. I believe that writing is the highest form of thoughtful expression, for in the fine word choice and editing; working to make everything perfect truly is what creates marvelous love letters.