I'm not sure the authors are examining the right variable. Very high individual or household income is actually a predictor of higher fertility than average, even in the developed world. Moreover, the fertility vs. income curve in the US at least is actually U-shaped, with higher values at both ends, but a depression in the middle.

It is only really the case that higher income suppresses fertility when talking about average income over time for a country or other large place. Nevertheless, that trend is quite robust, even though zooming in reveals other things happening too. This is a classic statistical paradox of aggregation.

I would never have used income as a metric to correlate or predict fertility; it's completely preposterous.

I believe that education is an indicator, because education in the developed world means preparation for a career and skilled employment outside the home. Therefore, a conventional education is to the detriment of home-making and child-bearing skills. An educated young adult would much rather start a career, and honestly doesn't know how to marry or have children! Some people, though, seem to frame it as "smart enough to avoid having kids" or "enlightened people know that large families are uncool".

Factors I could imagine affecting fertility: the security/permanence of marriage and family; the view of offspring as an asset and investment vs. a burden, expense, and liability; the demand for more than one parent to work/labor outside the home or be completely away from the family (such as war, exploration, travels.) The ability of children to contribute to household income and upkeep (Farmers and shepherds love extra unskilled hands!) Loyalty of children to accept personal responsibility, when parents are elderly and need care of their own.

Unfortunately it seems like all the above factors are conspiring against fertility at this time, in many areas of the developed world. No wonder immigration is such a flashpoint for us.