As I recall it became quite common after the Dems went nuts over a W campaign ad - the word "Democrats" moved through the shot and the left claimed the "rat" part of the word was some kind of subliminal hate mongering. Of course no one else had noticed it but they made such a stink about it we all started calling them "rats" - we figured, okay, if they're going to insist a 5 millisecond appearance of the letters "r-a-t" was a reference to them, we'd go with it.
That's my memory, anyway--others may recall earlier references but if so it wasn't common usage as it is now.