"Well, I know it sounds absurd that you'd be able to take a single object, disassemble it, and reassemble it to form two of the same object, but in fact it has been proven to be possible given infinite divisibility and something called the Axiom of Choice. If you're not familiar with that, I'd suggest reading a bit about set theory."

I actually can give you an "intuitive" justification of the Banach-Tarski theorem.

Suppose you have a rigid ball full of air. If you take half the air out and put it into another, identical ball, you now have twice the volume of air, at half the density. However, the points in a mathematical ball are infinitely dense - half of infinity is still infinity, so it turns out that if you do it just right, you can take out "half" of the points from a mathematical ball, put it inside another one, and end up with two balls that are both "completely full" and identical to the original one.