My brain hurts when thinking about that. How could 512 bytes be enough to store ~3 million bytes? I know that compression is basically about finding patterns and this sequences should be very compressible.
This is not exactly what happens here, but many compression algorithms work by recognising that certain substrings are very common, and give them a "shorter code". In this game, there are some quite long such strings, giving a good compression rate. Furthermore, because of the recursive nature, it can find such patterns again after the common substrings are replaced by shorter codes, because these codes again form patterns with repeated substrings. This goes on until there is almost just a bit of meta data and an "ur-pattern".
Compression is fascinating in many ways. For instance, since there are a fixed number of files of a certain size and some bigger files are made smaller, some smaller files must be made bigger by the compression! Of course, this could be as simple as attaching a header or flag which says "I could not compress this. Here is the old file verbatim." But that is still a bit longer than the original!
Precisely computing the Kolmogorov complexity of a given string could be very difficult, though. In general, it is uncomputable because we cannot decide if a given program will output a given string.