@jerry I haven't read that particular one but I have visited the Hilbert Hotel a few times!
The best way I've found to describe it is that comparing infinities is all about creating mappings, and there's an easy mapping from all rational numbers (that can be represented as a/b) into the Hotel. But an irrational number like pi can't map in -- it can't be represented as a/b (things like 22/7 are just approximations, and you never get pi exactly regardless of what large numbers you pick for the fraction).
So there's no room at the hotel for poor pi. And if you agree pi can't map in, then pi+1, pi+2, pi+3, and 2pi and 3pi, are also all irrational -- an infinite number of them -- and they also can't map in. And there's an infinite number of those little irrational bastards, so you basically have infinity-squared (aleph-1) trying to fit into infinity (aleph-0), and it just doesn't.
Which is a long setup for the joke:
Shit just got "real". :)