http://en.wikipedia.org/wiki/List_of_sovereign_states_and_de...
Has a lot of similarities to a per capita GDP map:
http://en.wikipedia.org/wiki/List_of_countries_by_GDP_(PPP)_...
The stronger your country's economy, the fewer babies you have.
While the naming is amusing in this case, it's a technical term that simply means that that, as your income rises, your propensity to have children decreases)[1].
The cookie-cutter explanation for this is that the opportunity cost for having a child is very high if your income is high; the opportunity cost of having children (particularly additional children beyond the first) is very low if your income isn't so high to begin with (and even lower if you make the assumption that lower income → lower costs for education, etc.).
[0] Here's an article providing another explanation, though the "standard" belief is that children have a negative income-elasticity http://econweb.tamu.edu/jlindo/AreChildrenReallyInferiorGood...
You have to be careful about causation and correlation there. It should be possible to curb population growth even before you get high levels of economic development by educating girls, rather than waiting for economic development to bring empowerment for women to decline to have lots of children.
It's a nuanced issue to be sure.