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.
A good way to develop an intuition for the fourier space is to look at simple images and their DFT transforms: http://web.cs.wpi.edu/~emmanuel/courses/cs545/S14/slides/lec... (3/4 of the way through the slide deck).
This analysis of a "bell pepper" image and its transform is also helpful: https://books.google.com/books?id=6TOUgytafmQC&pg=PA116&lpg=....
As for why you want to do this: throwing away bits in the spatial domain eliminates distinctions between similar intensities, making things look blocky. In the frequency domain, however, you can throw away high-frequency information, which tends to soften patterns like the speaker grills in the MBP image that the human eye isn't that sensitive to to begin with.
Or in this case, a real data set.
Along with the Wikipedia article and the obvious Internet search, there's a lot of good stuff that has been on HN: https://hn.algolia.com/?query=fourier%20transform&sort=byPop...
The JPEG article also has a very good, step by step example of the DCT, followed by quantization and entropy coding: https://en.wikipedia.org/wiki/JPEG
(Though for images it's in 2D, not 1D which is more commonly done)