So far as I can tell, without a graduate degree in physics, that book is a giant bucket full of minced-up grad-level math, and Penrose is a pretty terrible guide to it.
On the up side it made me think I should put some time and effort into understanding the math and the ideas. That was nearly fifteen years ago now, and it's turned out to be a lot of fun as an occasional side hobby.
On the down side, he completely fails to make even relatively simple concepts either simple or accessible.
When you spend your life surrounded by some of the smartest grad students in the world, you're going to have a very unrealistic idea of what the average human can understand. Of of how to explain things to the average human in a way that makes understanding more likely.
Infuriating book - but I'm still glad he wrote it, and that I tried to read it.
I will admit to getting a bit bamboozled quite often ...