Roughly speaking the problem is that mathematicians cannot come up with a meaningful definition of volume that applies to all sets of points (when I say cannot, I mean literally impossible, not just that they tried really hard then gave up). Instead, we have a definition that applies to a very large collection of sets of points, but not all of them.

Sets from that collection have a well defined volume, and any transformation which always leaves this unchanged is called volume preserving.

Sets from outside it, which the sets in the Banach Tarski paradox are, don't have a defined volume at all, and thus can interact with volume-preserving transformations in all sorts of weird ways.