I've always explained this as so (I have no evidence for any of this): 1. People in different social classes have different inherent desires in life. Some people simply want a family and have no other ambitions. 2. Lower class people (or people that have more children on average) have a harder time delaying gratification on average. Delaying gratification is a skill very beneficial when attempting to be successful and is related to the ability to think ahead and visualize the future. Having this ability instills different desires in a person and may detract a person away from parenthood due to truly understanding how much of a commitment children actually are (20+ years of your life, hours a day). Having this ability also helps one realize the difficulty in caring for many children.
Which is why development reduces fertility; people have higher confidence that their brood will survive into adulthood even if it's smaller.
(I interpreted the title initially to mean that rich people were having more children.)