Most likely, the factors which have contributed the most to a sustained demographic growth have been three: steady availability of nutrition at cheap prices, access to clean water, ever improving medicines and medical practices.
Breaking them down one by one, food is a staple which cannot be ignored. Aside from the obvious, (you need to drink and eat to survive), a constant flow of basic commodities at cheap, affordable prices is a huge boost in population growth. We can take as an example the Late Roman Republic and Early Roman Empire, where food was easily accessbile in urban areas for a price which was really affordable for most people, where it was not outright free to a select part of the city population (roughly 200.000 to 300.000). And if we compare the average cost of food during the Late Middle Ages in an important city of the Kingdom of Naples such as Capua, in 1477, and the daily wage of a labourer, a pound of bread costed roughly 20-40% of his daily income, whereas buying a loaf of bread toady is really, really cheap. We must also take into account the resilience and yield of modern crops, which are virtually granted to provided many hundred times the amount of seedlings used to start the cultivation with, against the hypothesized yield of 1-3 grains for each seed planted during the VIII century.
Freshwater availability again is both mandatory and auxiliary for survival and growth. Hygiene was not at the same level we expect today, granted that civilized people would wash rather regularly as no one likes stinky armpits and moldy hair for a month. It was however a vector for disease and infection, since there was no knowledge of water borne pathogens and sewage systems were incredibily rare during the Middle Ages. Thus, clean water would dramatically reduce epidemics of cholera, dysentery and other diseases with a very high infant mortality. This would partially have been solved by consuming alcoholic beverages of various strength even at an early age, if a fresh source of water was not available.
Medicine and medical practices (and pharmaceutics along with it) have literally saved mankind from centuries of suffering. The capacity to heal single or multiple ailments which would weaken and/or eventually kill a person without treatment, no matter how minor it would have been, made our life expectancy skyrocket. Reliable and constant information about a fetus' health increased the survivability of both mother and child during childbirth, vaccines (and that's a fact) eliminated lethal diseases which had suppressed constant demographic growth, like measles, tuberculosis, and smallpox. Antibiotics and germ theory saved countless lives which did indeed end centuries prior because a wound in a non lethal area would fester with bacteria and cause a septic shock. As recently as 1916 infected wounds would kill thousands because penicillin would be discovered in 1924.
Cherry on top and included in all of the above, is the constant development of technology which has allowed our species to keep extracting and processing more resources in order to increase our quality of life.
Of course, there several more factors (welfare programs, enlargement of markets, using more efficient and non-toxic materials, etc.) but I reckon that these few are at the base of most instances of sustained demographic growth.