Informally, you could say it is the AI "Rage-O-Meter" (patent pending), though that does not quite cover everything that the AI progress conceptually represents. A closer explanation of what AI progress represents is how much the AI considers the humans as a threat.
Anyways, the AI progress influences wave sizes, reinforcement sizes (or was it reinforcement rates?), and (at certain AI progresses), the tech level of waves. Also, at certain AI progresses, the AI will get a new bonus ship unlocked.
A few minor factions, including, but not limited to, neinzul enclaves, are influenced by AI progress, becoming more frequent and/or more nasty as AI progress goes up.
This may not seem like much, but wave size plays a BIG role in how much of a chance you have winning the game. Even if you are good at taking out a large number of ships, at high AI progresses the AI planets will get EXTREMELY well defended, making it really hard to get anywhere.
This is what I have seen so far. Let me know if I am missing anything.