The entire thing hinges upon the fact that the set is defined to be infinitely dense, and that if you divide infinity by two (or, really, more), you will still

The entire thing hinges upon the fact that the set is defined to be infinitely dense, and that if you divide infinity by two (or, really, more), you will still have infinity - thus, if it is possible to subdivide a sphere into two sets with the same radius, and the original sphere was infinitely dense, the two resulting spheres will have the same radius, and be infinitely dense - and thus pretty much identicle to the original. The rest is just discussing various other aspects of it, so that their structure is also the same. As for your grand-mother's story, I don't have a clue either - my best guess is that it's something about love not caring about how little sense something makes, but that doesn't entirely make sense.