The core of the matter is in the word "cutting". It is possible to pick points off of a connected sphere in such a way that you have 5 different collections of points on the surface of the original sphere (they are NOT connected surfaces). You then take these 5 sets, apply rotations and translations, and at the end, you have 2 spheres of identical volume. During the process, no 2 points from the same piece have changed their relative positions (of course, points from different pieces have).

The 5 pieces are highly un-intuitive objects. I imagine if you were to realize even an approximation of them, they would each look like a complete sphere, with infinitely many infinitesimally small holes poked through. But when you do put them together, each and every one of the (uncountably) infinitely many holes in the 1 sphere is perfectly plugged by one of the (uncountably) infinitely many points in one of the other spheres.