This isn't surprising at all to economists; we've known for ages that, on a micro (individual) level, children are an inferior good[0].
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.).
> The stronger your country's economy, the fewer babies you have.
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.
Conversely the first thing that happens when you educate women is they tend to enter the workforce on their own, at which point they gain better access to the means to control their own fertility.
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.