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.)

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.)

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."