It's not a great explanation b/c it doesn't go into what "measure" and "volume" mean.
Tough to do without going through a full course but a slightly-less-handwavy explanation would go like this:
There's a mathematical tool (called measure) that formalizes-and-generalizes the notion of "volume".
It generally behaves very much as you'd expect; your intuitions about how "volumes" combine and intersect will generally apply.
There's a catch, though: the way the tool is constructed leaves open the possibility of a non-measurable set, meaning a set for which the definition of the tool leaves you unable to assign that set a well-defined "volume"; you can't just assume that the measure of a set exists.
If you assume the Axiom of Choice then not only are such sets possible, but you can construct non-measurable sets.
The core process in Banach-Tarksi looks like this:
(1) take the sphere (a 'nice' set, which we'll say has "volume" V)
(2) divide that sphere into some sub-sets that are non-measurable (are sets for which our tool cannot supply a measurement)
(2.a) Effect on total volume: should have no impact, as the parts we have reassemble to an object of known volume
(3) move the subsets around by sliding-and-rotating them
(3.a) Effect on total volume: should have no impact, as neither sliding nor rotating changes volume
(4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)???
The paradox comes from getting double the volume through a sequence of operations that are apparently volume-conserving.
You can go with this a couple different directions.
I'm not convinced this should be an intuitive outcome.
An intuitive, hand-wavy explanation for what's "really going on" would be something like non-measurable sets carry around infinite amounts of finely-detailed structure (too finely-detailed to perceive using our measuring tool); depending on how you position some sets relative to each other their finely-detailed structure might either cancel out (adding no volume) or reinforce each other (adding lots of volume).
That said I think the real lesson here is that your intuition is trying to have its cake -- a non-measurable set -- and eat it too -- have the "volume" of a non-measurable set be preserved under volume-preserving operations.
