I always feel part of the confusion with Banach-Tarski is that lots of words don't use their "natural definitions", which makes the proof more surprising. People (not this article) often talking about "cutting" a sphere, which is really misleading.

This result is, in many ways, quite similar to the idea I can "cut" the integers into the odd integers and even integers (but with many more fine details).

This is still a nice article, which explains the actual result well.