I see from the comments on YouTube it's a variation on a Yiddish song complaining about having only potatoes all the time.
When the Fugs version starts out, it reminds me a little bit of the Toy Doll's version of Nellie the Elephant: a rock band subverting a kids' song.
The Fugs go way, way further, gradually building to a climax of nihilism. The contrast between the relatively familiar and harmless beginning and the surrealistic end, with no clear seam or border between them, is funny, in the way that Monty Python's Cheese Shop sketch was funny.
That was my take. But then I thought Waiting for Godot and a lot of Franz Kafka was funny, too.
I think Waiting for Godot is hilarious and but that the song is not.
and get nothing!
Seriously, this strikes me and a lot of people as pretty funny, but isn't much like the typical joke.
Edited to add: Poll and discussion. Short version: being amused by the song is rarer than I would have thought, though the LW consensus may be usually strong.