Everything he said.
A "people pleaser" is someone who tries to make everyone else happy all of the time.
"compulsion" is an emotion. If you feel a compulsion to do something (compelled to do something) you feel like you must do it.
In English when we say we "quiet" an emotion it just means we make it less strong. Ex: "He learned to quiet his anger"
So "a quieting of the compulsion to be a people pleaser" just means the person felt less compelled to be a people pleaser.