I don't believe I have to spell this out...
For a population to remain stable the birth rate has to be the same as the death rate.
Historically, the combination of high infant mortality and poor health care generally meant that families needed to produce anywhere between three and five children just to keep numbers steady.
Advances in health care generally and perinatal care specifically meant that from the late 19th century onwards fewer and fewer children are required to maintain the population size (typically now it's 2.1 children per couple).
However, up until the 1960s it was difficult for women to control their own fertility and so birth rates were still relatively high - hence the huge growth in population of the 1930s onwards (the baby boom wasn't so much due to an increase in birth rates, but due to more babies surviving).
Throw in the fact that life expectancy increased by 50% or more since the late 1800s and low birth rates aren't a problem, they are a
necessary requirement if we are to avoid the worst effects of overpopulation. The world population took 123 years to grow from 1B to 2B, and only 85 years to go from 2B to 7B.
The developing world is still on the front side of these population demographic curves - the green revolution allowed them to support larger and larger populations on the same amount of arable land which has lead to surges in populations. However, increased affluence is leading to similar cultural changes as we experienced in the west - more women are seeking education and controlling their fertility as a way out of being baby-making machines. Over the next couple of decades we will most likely see their birth rates dropping in a similar fashion to how they have in the West.
Edit: Just wanted to add the chart below which makes the rate of population growth over the last century quite clear: