The rapid increase in world population during the twentieth century is primarily explained by the fact that death rates decreased rapidly while birth rates remained relatively high, especially in developing countries.
Advances in medicine (antibiotics, vaccines, surgical techniques), improved sanitation and clean water supply, better nutrition, and public health campaigns dramatically reduced mortality from infectious diseases, infant mortality, and maternal mortality. Life expectancy increased substantially across the globe. This meant that more people survived to reproductive age and lived longer, leading to rapid population growth.
This phenomenon is described in the demographic transition model: Stage 2 of the model shows death rates falling sharply while birth rates remain high, producing a large natural increase in population.
Increasing food production helped sustain the growing population but was not the primary demographic driver. The abolition of the slave trade occurred in the 19th century and did not directly cause twentieth-century population growth. Birth rates have actually declined in most parts of the world during the twentieth century, not increased; it was the gap between still-high birth rates and rapidly falling death rates that caused the population surge.