Oh Richard Feynman, the patron saint of scientists, engineers and nerds everywhere. Richard Feynman was a Manhattan project physicist during WWII, and later went on to teach and popularize physics. He was a weird, engaging character whose writing is some of the best on physics topics. What Do You Care What Other People Think? is random memoirs, written casually and charmingly along with a look at the Challenger space shuttle disaster.
If you are looking to read a biography of Feynman, or about his work on the Manhattan project this is not the right book. It is also not about science, in general, and won't teach you anything about physics. This book contains several shortish anecdotes Feynman recalls from his childhood on. They really are fairly random, and sort of highlight how Feynman thought and how much he thought of himself. Not that that is a bad thing when you are Richard Feynman!
Almost half the book is devoted to Feynman's role in a commission looking into the space shuttle Challenger disaster, which occurred 11 days after I was born, in January 1986. Feynman worked on the commission, was sort of a loose cannon as far as the commission's work went, and ended up writing his own little report on the disaster.
For me the most interesting (and depressing) part of this book was the ineptitude of NASA Feynman highlights in his Challenger section of the book. According to his research and interviews, NASA's engineers and astronauts were amazing, dedicated individuals whose insight into problems the shuttles may have been having was squashed, discouraged and hidden by NASA's management. Really depressing. I have no idea how contemporary NASA compares to late 1980s NASA, but I seriously hope that they took Feynman's words to heart and shaped up.
In summary, this book is equally divided into two halves containing charming anecdotes about how awesome Richard Feynman was and Feynman's analysis of the Challenger disaster. If you are interested in physics, NASA, or Feynman specifically this is a book you will probably enjoy. Otherwise, maybe not so relevant.