That's because you either have a tab open with a page that has a chat box on it (such as the leaderboard page or the bounty page) or you are idling in the chat room.

I've been told that also, but I still don't get logged out

There, mystery solved! When you first logged in, you must have checked the "keep me logged in" checkbox.

No, no mystery solved yet.

You CANNOT stay online unless *something* is accessing the website with a session that is linked to your account.

Meaning, something on your computer is talking to the game's server. If it isn't, someone else is