Here's the version I always heard growing up:
Bud Abbott: Strange names. Pet names, like Dizzy Dean
Lou Costello: His brother Daffy
Abbott: Daffy Dean
Costello: And their French cousin (or maybe he says "I'm their French cousin."?)
Abbott: French?
Costello: Goofé
Abbott: Goofé Dean! Oh I see
[audience laughs]
The audience thinks it is funny, but I never got the joke. Does Goofé have some meaning that we don't get now, or is it just a nonsense joke?
While looking for a video to link, I found many versions where they don't use this joke, and 2 where they use this variant:
Bud Abbott: Strange Names, like Dizzy Dean
Lou Costello: Brother Daffy
Abbott: Daffy Dean
Costello: I'm their cousin
Abbott: Who are you?
Costello: Goofy
Abbott: Goofy? Goofy!
Maybe it is a reference to Goofy meaning a left-handed player?
I just don't get it.
The Dean brothers were good baseball pitchers in the 1930s, especially Dizzy. This was a period in which baseball was an important element of popular culture so just about everyone knew who they were and Dizzy was also known as fun-loving kind of guy.
The adjectives dizzy and daffy are similar in that they each suggest a kind of off-kilter state of being or, applied to the brothers in this case, as descriptors of slightly off-kilter behavior. But not in a mean way. The adjective goofy suggests the same connotations. So, the joke here is that the imaginary third person (the cousin nicknamed Goofy) is supposedly cut from the same cloth. Lou Costello was also known as a lovable, off-center person. Lastly, and simply another cultural reference of this time, the Disney character, Goofy, was also popular in the 30s. Add all these predicates together and you get a pretty good joke which a lot of people would understand and appreciate.