The key principle is simple.

If you want to best predict what state comes next from a space of possibilities, you have to figure out how probable each next state is and pick the most probable one.

If you want to compress something, you have to figure out how probable each next next state is and assign the smallest code to the most probable state.

These two processes are essentially the same, and the resulting structure of a system which regulates either process will be similar, approaching the same structure at high confidence.