It's not actually a clear inverse relationship on the individual level, even if the data shows a correlation at the national level.

There are a few things happening that complicate the analysis at the individual level, too:

• Wealth/income are correlated with age, and 40 year olds tend to have both higher incomes and lower fertility rates than 25 year olds.
• Wealth also correlates with race, for better or for worse, and there have always been persistent differences in birth rates by race.
• The sample sizes aren't big enough to show whether the very rich (95th+ percentile) actually reverse the trend, to where being richer is correlated with higher birth rates, where the curve ticks back upward at very high incomes.
• The correlation is actually the other direction when looking at the individual incomes in certain countries (Netherlands, Sweden, Norway), and the effect is stronger when looking at men and their incomes.

Other country level data also suggest that there are big cultural factors in birth rates as well.

All in all, the relationship between income and fertility is complicated, with lots of other factors at play.