Can someone explain how the frequency domain stuff works? I've never really understood that, and the article just waves it away with saying it's like converting from binary to hex.
In the most basic terms, not even talking about frequency, the mechanics of this is that one series of numbers (pixel values, audio samples, etc.) is replaced, according to some recipe or another, with a different series of numbers from which the original can be recovered (using a similar "inverse" recipe). The benefit of doing this comes from the discovery that this new series has more redundancy in it and can be compressed more efficiently than the original, and even if some of the data are thrown away at this point, the purpose of which is to make compression even more effective, the original can still be recovered with high fidelity.