Technically it is self-contained and nothing beyond high school knowledge is needed to understand it; but while technically true, in reality it's a cruel deceit. If you try to learn it without advanced training in physics and math, you'll run into limitations related to mathematical maturity (https://en.wikipedia.org/wiki/Mathematical_maturity) very quickly, within the space of the first few chapters.
The idea of this book seems noble and uplifting, but it cannot be done in this fashion. It doesn't matter that every theorem and assumption are clearly stated and follow each other in an orderly sequence. There's a huge number of new ideas, mathematical and physical, which require proper acquaintance to settle in your mind, not the 1-2 pages Penrose is able to give each of them. If someone is serious about learning physics to that level "properly" and is willing to invest the time and effort, they're much, much better off working through conventional college-level (and later grad-level) textbooks in particular subjects, taking care of math prerequisites before and during such study. That's very very hard to do, but is in fact possible, whereas studying solely from Penrose's book, I think, is not. It functions, sadly, as a sort of trap for impressionable bright people very eager for advanced knowledge - they will try to struggle through it, inevitably give up rather early on, and blame themselves.