Took me awhile to figure this one out, but once I did, life was significantly better.
The bottom line of the book is that people that want to get along fine with others should be wary of the other people's emotions. For instance, of the the first "principles" in the book is to not criticise, condemn or complain people openly, because people get defensive (an emotional reactional) when you do that, and you can't get nothing out of them. The rest of the chapters are also related with other people's emotions.
As for "genuinely cares", the book also talks about it in some chapters. What I could get out of it so far is that someone who listens with attention is mostly grasped as someone who genuinely cares about other people..
All in all, the bottom line of the book is all about "managing" the other people's emotional reactions..