Why are birth rates so high in these regions and not in more developed areas?

Development is exactly what makes birth rates go down. People also used to have more kids in developed countries in the past, and it went down the more the countries' development indicators went up.

The reason? It's not simple, there are entire theses over the subject and, likely, different causes, but roughly, it's lower education levels, more religious influence, less women rights, criminality, etc. For many people in poorer places, having kids is not an option at all. It "just happens." Actually choosing to have kids is something very recent historically and far from something universal.