I've heard from a friend who writes/reads stories on the site Fanfiction, and she says that some of the site developers want to clean up the site. Clean up meaning--> removing stories that include violence, sexual situations, etc. But isn't that kind of restricting for the writers, if those details happen to be in their writing? The motto of the site is "Unleash Your Imagination", so wouldn't this be going against that? And if it's to protect the younger readers/writers, shouldn't they know what to and not to read/write? Any thoughts?

Doesn't matter. Their site. Their rules. People getting butthurt over it, won't change it.
We, the people, cannot do anything to just magically change their opinion. You can try, in hopes of doing it, but there is no fact itself, that will state you can stop it.

So unless everyone is willing to try, and possibly fail, they need to stop crying over it. They need to back up all their stories before uploading them onto the net. It is their fault if they lose their work. Not the people who own the site's fault.

Again, they can do whatever the Hell they want to their site. They could ban someone just for shits and giggles, let alone deleting all their work for no reason if they wanted, and nothing we can do, will ever change it.