If you’re receiving Direct Messages from yourself that you didn’t send, chances are you signed up for an RSS feed with your Twitter info.

If you don’t want those messages any more, just change your password! If that doesn’t seem to be the issue, go ahead and file a support ticket.

Emphasis mine. Here’s the core reason why application keys are useful: Control. With an app key, the user (or Twitter) could just have blocked the offending app. Instead individual users have to update their password on every single Twitter app and service they use.