2) The word you are asking about is "falter," not "flatter." They are completely different.
To "falter" means to stumble, to hesitate for a second. It suggests a momentary difficulty or mistake. I think Sacks is referring to Temple Grandin's talking, not her driving. She is speaking fluidly, smoothly, eloquently, and then she "falters"--stops speaking. In this case, she stops speaking for a moment. Her speech falters. The reason is that she has started to weep.
3) To "flatter" somebody is to say nice things about them in order to please them or influence them. Flattery is often false. (We often use the word "insincere." Flattery is often insincere.) For example, a salesperson might say "I really like the way you look in your clothing, you're a very sharp dresser." He doesn't really think so. But he hopes it will make you feel good and buy more.