I don’t know what to make of this.
Letterman seems out of sorts, and the audience is in a weird mood, and most unlike him, Billy struggles to find the funny in a train of thought. That said, the weird vibe gives them both something to play with, and they do a good job of dealing with it.
The two related sitcoms that get mentioned are Head Of The Class and Billy. Here’s a clip from the first.
And here’s a weird, nonsensical trailer for the follow up, Billy.
Clement & La Frenais wrote some episodes of this.