This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
The Book of Proof teaches various methods of proof and how to use them in Mathematical statements. The section on the fundamentals was beneficial since I tend to forget the Set Theory symbols. I haven't had an official course in Set Theory, so I need to take notes on it or devote it to memory.
So Richard Hammack wrote this textbook for his students. It explains the theorems and ideas admirably. If you are learning on your own, like I am, this textbook is excellent.