AP Human Geography time!
There are five stages of something called the demographic transition:
Stage 1: high death rate and high birth rate, low population growth
Stage 2: death rate declining do to innovation, but still a high birth rate, large population growth
Stage 3: low death rate, birth rate declining due to socio-economic reasons, some population growth
Stage 4: low death rate and low birth rate, little population growth
Stage 5: low death rate, but birth rate is still going down, possibly negative population growth
Until a few hundred years ago, most of the undeveloped countries were still in stage one of demographic transition. They had very little in terms of medical technology, which means that people died young. The people also had a lot of children, for many reasons. Children supply low-cost labour on farms, giving extra income to the family. Having a lot of children also countered the high infant mortality rate. A lot of children died, so you need to have a lot of children to make sure that several survived. A third reason is tradition, as people tend to want to have large families if everyone else has had large families.
Also a few hundred years ago, western civilization was already well into stage 3 of the demographic transition. Centuries of slow and steady scientific progress allowed the death rate to drop gradually. Also, socio-economic pressures slowly reduced the birth rate. People were now longer working mostly on farms, children became more costly and not worth the time, and eventually the tradition of large families was starting to go away. Since both of these processes were gradual, a large population explosion was avoided.
In modern times, all developed countries are well into stage 4, and a few are in stage 5 if you discount immigration. Until recently most undeveloped countries were still in stage 1 or 2. Then, advances in technology had allowed the world to become a much more connected place. This is called “space-time compression” in Human Geography jargon.
Suddenly, all of the medical technology of the developed nations had been introduced to the undeveloped countries. This extremely fast pace of development caused a tremendous change in lifespan, allowing to live much longer. The death rate plummeted. The infant mortality rate was also reduced, so more children were surviving to adulthood. The birth rate, however, is much slower to change. In many rural areas in developing countries, it is still economically advantageous (at least not disadvantageous) to have more children, even though there is a lower availability of food because of the extra children. There is also a long transition time, because there is not going to be an instant cultural shift in birth rates.
This combination of factors (a dramatically reduced birth rate and a birth rate that is still very high) has resulted in the massive population boom in developing countries that we are currently experiencing.