What are two reasons why the world's population has fluctuated throughout history?

Two major causes in population fluctuation could be described as reproductive capability and environmental pressures. Both of these are related to access to resources, which I will address further in just a moment.
First, humans have a somewhat limited reproductive capability. Though we differ from other mammals in the sense that women are able to conceive year-round during their fertile years (rather than seasonally), we must account for a gestational period as well as time needed to nurture a newborn. Prior to the last century or so, it was expected the world over that at least one, if not most, of a woman's children would die. Many couples tried to maximize their success by having many children with the expectation that some would die, but they had to be able to provide for the children from then on in terms of food and shelter. With the technological advances we have today, it is quite unusual for a child to die, so many parents choose to have less than five children. (Though it is technically possible to have many more over a lifetime!) 
In the long history of human reproduction, populations generally remained stable or grew slowly, as a couple might have a small number of children who survive to adulthood. The slow growth that did occur was the result of parents having at least two children outlive them. Periods of rapid population growth were unusual for much of history because it creates a huge increase in demand for resources like space, food, and water. Such rapid growth would be unsustainable if a population had a long history of stability or slow growth. Imagine if a couple had two children and made enough money to provide food and a comfortable home for the four individuals. Then, the parents have twenty more babies but do not move to a bigger home or increase their grocery spending. This is a somewhat reductive example, but on a macro scale this is both unlikely and unstable. When it comes to population dynamics, the fertility of adults isn't enough for population growth—there must also be sufficient resources to sustain new individuals.
Of course, populations can decline, too! Disease and famine are primarily responsible for large population declines. We can also consider natural disasters and human activity, but the environment plays a huge part in population fluctuation. For example, the Medieval Warm Period created agricultural stress the world over. This was enough to cause famine in many parts of the globe, and in some places, resource competition was so high that violence increased dramatically. The combination of widespread starvation, infectious disease (which thrives in warm temperatures), and violent conflict greatly reduced global populations.
https://governmentshutdown.noaa.gov/

https://www.britannica.com/science/population-ecology