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Nisan comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion
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I like these. They are both perfectly correct, and clearly express important and surprising ideas.
They give you an understanding of parts of Gödel's theorem and the Banach-Tarski theorem, but not the whole thing. Mark Dominus' caveat does a good job of indicating the gap between Raymond Smullyan's explanation and Gödel's theorem. There's also a gap between MarkCC's explanation and the Banach-Tarski theorem.
MarkCC puts the natural numbers into one-to-one correspondence with two copies of the natural numbers. Similarly, the points in a ball can be put in one-to-one correspondence with the points in two balls, although it's not immediately obvious. It's easier to see how to put the ball in one-to-one correspondence with a ball of twice the diameter: Just expand the ball. But even knowing this, one might assume that you can't increase the volume of the ball without stretching it somehow. The Banach-Tarski theorem (which requires a lengthier explanation) indicates a way to do just that.
I would say that the entire value and interest of the Banach Tarski paradox lies in the fact that you are restricted to seemingly volume preserving transformations, thus proving that volume cannot be meaningfully defined on all sets of points in space. This is not an obvious fact, and in fact we can come up with definitions of volume that work on very large collections of sets of points in space. Without that you just get some very basic set theory, which is a lot less interesting and surprising than Banach Tarski. I wish the author had only claimed to be explaining basic set theory, instead of explaining Banach Tarski, in which case it would have been quite a good explanation.