Nowadays they shoot the original characters against a blue or green screen and then generate the background either by a different film or by computer. The Jim Henson Hour was shot that way with Jim and the Lion in the beginning, and the Muppet Central part of the show was shot against a blue-screen (I believe) and the computers and tv's were computer generated later.
In the old days, the puppeteers wore black with a black background and shot the film and then overlaid this to background film, and that is why you can sometimes see through "Disco Frog" or "Octopus' Garden" Piggy and Kermit to the background. Also, if you look REALLY close, you can see a "ghost" of the performer, too!