There is something called "demographic transition": birth rates take some time to adequate to reduced mortality rate. For one generation at least, people keep reproducing as if mortality was still high, and population booms. Yes, it happened in Europe, in the late 19th century - which helps to explain the enormous European emigration to the "New World". It happened in Latin America in the second half of the 20th century. And it is happening in Africa now. So, cultural reasons, yes - but not cultural reasons that oppose Africa to Europe; cultural reasons that have to do with people being accustomed to have many children, otherwise none would survive to adult age, and taking some time to realize that this is no longer necessary.
Why do mortality rates fall?
Better sanitation, eradication of marshes and swamps, vaccines, antibiotics, better techniques to fight infant mortality, especially diarrhea, better understanding of why and how diseases spread.
Why do birth rates fall?
Contraceptives and pension systems (that make people able to maintain themselves in old age, and unnecessary to have an nth child just to take care of them at the end of their lives).
In the specific case of Africa, there is also the fact that its population growth had been depressed for some three centuries by the slave trade. When the slave trade was abolished, in the mid 19th century, the population obviously rebounded.
Malaria was an endemic problem in Europe during centuries, up to WWII; its name comes directly from Latin not because Latin was the scientific language of European scientists from Renaissance on, but because Latin was the widespread language of the Roman Empire. So, while it is true that malaria originated in Sub-Saharan Africa, and is a "tropical disease" in that sense, it is not true that malaria is a "tropical disease" in the sense that temperate regions would be naturally protected from it due to a climate barrier.