Intellectually poor man's attempt at a summary of the summary:

Part 1 - math proof that a formal system when evaluated by an agent "breaking out of the system" can use that system to prove itself inconsistent. ("f* up" the internal logic of that system by feeding it into itself)

Part 2 - ironically, to be conscious you have to have an awareness of your own system (a "strange loop") but you'll never be able to understand yourself as proven in part 1.