It is obvious that Granny is trying to tell a joke here — and failing miserably. The problem was that quite a few readers (including yours truly) were having trouble figuring out what that joke was supposed to be in the first place.
People started asking about the Alligator Joke so frequently on alt.fan.pratchett, that eventually Terry himself posted the following “definitive explanation of the alligator joke”:
“It is (I hope) obvious that Granny Weatherwax has absolutely no sense of humour but she has, as it were, heard about it. She has no grasp of how or why jokes work — she’s one of those people who say “And then what happened?” after you’ve told them the punchline. She can vaguely remember the one-liner “Give me an alligator sandwich — and make it snappy!” but since she’s got no idea of why it’s even mildly amusing she gets confused… all that she can remember is that apparently the man wants it quickly.”
When conversation on the net then turned to the origins of the joke, he followed up with:
“As a matter of fact, I’m pretty sure I first came across the joke in an ancient US comedy routine — Durante or someone like him. It sounds burlesque.”