"As explained in the preface, the main prerequisite is some amount of mathematical maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on."
Yeah, that's way beyond what's called basic math instruction, e. g. in schools. A more specific, as in accurate, subtitle (or description) is in order.
Apparently the author tried to somewhat expand the audience from that, but to me it seems still mostly appropriate for smart high schoolers who have heard some pieces of lore from friends about these topics, but they can't put that puzzle in order in their minds yet.
It's most definitely not aimed at the average student. You need to be highly curious, motivated and find math fun already.
And I think that's a perfectly fine thing. It's great to have books for that kind of audience.
Higher mathematics isn't necessarily very strictly defined anyway, but I guess most people who've heard the term would apply it to branches of math that are developed using formal definitions and at least moderately rigorous proofs, and that usually aim at a level of generality beyond their originally motivating examples.
I'm not saying you're wrong, I know for a fact that you aren't: unfortunately most high-school students fall extremely short of that bar, but it's not necessarily that way. Many teenagers can and do develop that kind of mathematical maturity.
In this context "basic" means "it doesn't require knowledge in the field", and by and large this book can indeed be followed with no other requirement than the mathematical maturity it talks about. Many classic books self-describe in similar way.