Again, education of women.
In many of these cultures, women don't have the right to say no. The better educated and economically advanced women are, the more likely they are to be in a position to say no. Also if you have the option of feeding your children, or buying birth control, most will choose to feed their children.
Also international money for family planning programs are largely tied to politics in the countries sending the aid. The US has opted out of funding many international family planning programs because of political opposition to abortion education and abortion providing.
Children also provide an economic advantage is some countries. Sometimes the only way to add farm labor is to raise your own. And in many countries, people expect their children to care for them when they are old.
The other issue is lumping the whole "Third World" together in these conversations. The needs of Sub-Saharan Africa are very different from those on the Indian sub-continent or in South America. There is everything from rural standards for families, stanch Catholocism, China's one child policy and India's preferential abortions of girl fetuses with plenty of variations in between playing into issues of family planning.
Opps, that last sentence is kinda confusing. What I meant to say is....Family planning and birth control is fighting many ingrained and entrenched ideas including: the need for children to help in farming, religious beliefs, cultural practices, family status and many other factors.
Complex problems rarely have simple solutions.