This is not so much "how to do mathematics" (to quote the OP), but more how it feels to do mathematics. A New Yorker review said this book "is less about math than about mathematicians—how they live, how they work, and how they talk to one another." I found it inspiring.
Villani, Cédric. Birth of a Theorem: A Mathematical Adventure. Farrar, Straus and Giroux, 2015. (Guardian review. AMS review.)
[![Villani][1]][1]
The AMS review ends: "The implicit task that Villani had set for himself, of explaining what it is all about, is a difficult one .... To my mind the book succeeds wonderfully."