It occurred to me that a better way to approach explaining the Banach-Tarski paradox to a layman might be through the story of the infinite hotel. If you aren't familiar with how an infinite hotel that is full can still make room for an infinite number of guests, a good introduction is here: http://diveintomark.org/archives/2003/12/04/infinite-hotel
Let's assume you know and understand the infinite hotel thing from one of Martin Gardner's books, the link above or any other source. Visualize a number line with the points 1, 2, 3, 4, ... on it marked as "rooms". Now when you make room for just five new people, by moving existing guests 1->6, 2->7 and so on, and freeing rooms 1-5, you can look at it as shifting all the "room points" five units to the right. When you need to make room for an infinity of new people, and you move guests 1->2, 2->4, 3->6 and so on, this isn't a simple shift, because points move non-uniformly: the farther away, the farther you move. But it turns out that that's just because you don't have much freedom of movement, so to speak, in one dimension.
It's even more useful to look at the hotel process "in reverse": say you have all the rooms taken, now people in rooms 1, 3, 5, 7... all move out and renumber themselves, founding another hotel of the same kind, while people in rooms 2, 4, 6, 8... squeeze together, each moving to the room half their original number. So you start with one hotel and you get two identical ones, again with all the rooms taken. And again, the manner of movement here is non-uniform, but it's because one dimension is too crowded.
In two dimensions, there's a way to shift an infinity of points to become two identical infinities, but all the movement is simple shifting together or rotating together. You divide the points similarly to the even versus odd numbers division in the hotel example, but because you have a lot more space to move around in two dimensions, it turns out you can move all the "evens" and all the "odds" uniformly with respect to each other, as if you were shifting and rotating them together in the physical world. But the end result is the same: two infinities where one was, and the basic idea is just the one with the hotel rooms. The actual way you divide the points into two groups in two dimensions and shift/rotate them is the tricky technical part of the proof you'll have to take on faith here. It's not very complex math, but it does require some abstract higher math knowledge, at about the level of a math major college degree.
OK, so given all that, what do we do with a ball? In a ball, we first look at just its surface - the sphere - which is really two-dimensional. You can take an infinite mesh of points in two dimensions - the one we learned to "duplicate" with the hotel process above - and stretch them over the sphere, like a lattice. It's not too difficult to show that by wiggling around this lattice of points you can cover the whole sphere with its copies, and the tricky hotel-like rotation and shifting that you do with one lattice, you can do with all of them together in sync. So it looks like we are breaking the sphere down into two parts, and shifting/rotating them around to get two spheres next to each other. Each part is the composition of one half of the infinite lattice - one half of the "hotel rooms" in the two-dimensional hotel - collected over all the wiggled lattices together. Only it turns out to be more complex to unify them like that, so it requires four parts and not two.
Now, these four parts are unbelievably complex-looking. Just as with the original hotel puzzle there's a break with intuition where you get two of the same from one, here you do this infinitely many times at the same time, in two dimensions. Nothing like that could be done in the physical world. You're basically taking the sphere, breaking it down to individual points, and them juggling them very intricately hotel-like in an infinity of configurations together. The point is, any intuitive notion of "volume" or "space taken" by the sphere breaks down in this process, becomes irrelevant. With the infinite hotel, two hotels are also taking up twice more "space", but we don't perceive that as especially freaky on top of everything else, because they stretch to infinity anyway. But they don't have to; there's an infinity of points inside a fixed volume too. You could host the infinite hotel on a surface of a sphere if you were willing to make the rooms really tiny (one point each), and this is kinda what happens in the Banach-Tarski paradox. So the paradoxical sense of getting something from nothing is because in the physical world, we can never go to the scale of individual points, where the notion of "volume" loses relevance. But in math we can.
Well, back to the ball - if I convinced you, with lots of handwaving, that you can break down the sphere into four unbelievably complex-looking parts and reassemble them into two spheres, balls are now easy. Every time you do something to a point on the sphere, think of a ray from that point to the ball's center, and do exactly the same shifts and rotations to all the points on that ray. This way, you're sort of shifting and rotating many concentric spheres at the same time, all the way from a single point in the center to the surface of the ball. And each sphere gets reassembled into two of the same, so the entire ball gets reassembled into two of the same. You do need a bit of a special treatment for the very point in the center, and that's your fifth "part".