How have people caused overpopulation?

"Over population" means more individuals than can be supported by their environment; space, food-supply, shelter and so on.

Taken as a whole, the human population of about 7-billions does not exceed the planet's ability to provide space, food-supply, shelter and so on.
The main problem is one of distribution in both the human population and the necessary support-systems.
Every year in USA about $19-billion value of food is scrapped before harvest, or damaged in processing or else discarded when people are satisfied with their diet.
That volume of food could maintain a population of about 25.3-million in decent food for a whole year in poorer parts of the planet.

Also, in many parts of the planet, the human population needs to move to better environments; to gain access to more space, food-supply, shelter and so on. But politicians are preventing it; either by holding people almost as "prisoners" within a territory; or else preventing them from entering other territories because of the existence of imposed boundaries.

To answer your question in strict demographic-science terms, human breeding patterns have not changed very much over the past 200 years but the incidence of disease has fallen dramatically. Infant mortality across the planet today is about 3% of live births by the age of 5 years. In 1880, it was nearer 25%. In 1780, it was about 34%.
Also, the average age at death across the planet has risen from 43 years in 1900 to almost 70 years today.

In this sense, medical-science and dietetic-science together have produced a population explosion.

It would be unwise to reduce the birth-rate to less than 1.4 per capita [say average 2.8 children per couple], or else the human population could fall into the "Allee-trap." Unable to sustain its size because of the shortage of women of fertile ages.
There is an example of this in Western Europe today. The latest figures from the European Commission show that the average fertility rate in the European Union stands at 1.55 births per woman, but in some countries it is below average: 1.27 in Spain, 1.2 in Portugal, and 1.3 in Greece.
Total populations in these countries are falling.

Too many "birds and bees" activity. Plus, better and more food, better medicine, better sanitation, less war (for now).

By breeding too much and not sharing world's resources