3 ideas
Criticism is futile because it puts the other person on the defensive and makes them strive to justify themselves, wounding their sense of importance and arousing resentment rather than changing behaviour. Carnegie's evidence is that even criminals like 'Two Gun' Crowley and Al Capone saw themselves as misunderstood benefactors, so if the worst offenders don't blame themselves, ordinary people won't either.
The only way to get anyone to do anything is to make them want to do it, so every request should be stated in terms of what the other person wants rather than what you want. Carnegie's example is the father who cannot get his son to eat until he connects eating to the boy's desire to beat a bigger neighbourhood bully, which turns persuasion into finding the other person's existing motive and attaching your goal to it.
You cannot win an argument, because even if you prove your point the other person has lost face and now holds their view more firmly. To change a belief, Carnegie says to show respect for the other person's opinions, never say 'you're wrong,' admit your own errors quickly, begin with points of agreement to get early 'yes' responses, and let them feel the idea is their own, so that changing their mind does not cost them status.