In your choice, most in most of people's lives means something similar to the meaning of most in, say, Most of my life was spent in prison.
On the other hand, the proposed choice by the book means this:
More people think climate is important than there are people who don't.
In the case of the first exercise, you were wrong. I have to think about how I can explain the why's and wherefore's to you.
Student or Learner