What exactly does "The guy's multiplying faster than a Catholic rabbit." mean?

Rabbits multiply very quickly. There is a saying that some people "breed like rabbits" which means they produce too many babies very quickly.

As for the catholic component of the joke, that's because the Catholic Church proscribes any form of birth control, so a catholic couple is likely to have more children than non-catholic couples. Now, if your rabbits happen to be Roman Catholics (and of course that's a joke) there will be a tremendous baby boom on your farm.

• "breed like rabbits", definition - reproduce prolifically.

Edit - The truth, however, is that ever since the birth control pill and other contraceptive methods became available, a large number of observant Catholics also use birth control techniques.