Explanation:
The real disparity is between agrarian traditional societies and industrialized modernist societies. Only incidentally does that usually coincide with developing vs. developed societies, but there are many relatively recently industrialized societies that are found in emerging countries, usually with middle-tier economies and urbanized societies.
Having lots of children is an asset in a pre-modern society (or one just starting to modernize), because agrarian work is labor-intensive, and infant mortality was extremely high and made people have many children as a sort of safety measure, because some or even many of the children could die until adulthood or in their early adulthood even. Besides, traditional societies were usually more communitarian and had lower standards of quality of life, consumption and professional requirements, which means having children was less expensive. Urban environments usually also pressure people into having smaller families, because of housing constraints and the different needs to make a living in a city.
Several underdeveloped but not exactly poor countries, like Brazil, China, Iran, Lebanon and Thailand, usually underwent particularly a rapid (in comparison with developed countries) transition from the demographic patterns of traditional pre-modern societies to post-industrialization modern societies. Because of that very fast transition, which took much longer to happen in most “old” developed countries, they have slightly lower death rates, but often just as low birth and fertility rates. That means they have moderate population growth even after achieving sub-replacement levels, but they won’t for much longer.
