Wild ancestors of banana trees grew in southeast Asia. Like chickens, bananas were first domesticated in southeast Asia, possibly as early as 8000 BC but certainly before 5000 BC. That’s about the same time as apples and earlier than peaches and avocados and oranges. In southeast Asia and Papua New Guinea, there are dozens of different kinds of bananas or plantains (plantains are just a less sweet, starchier kind of banana).
Banana farming slowly spread west to India and from there across the ocean to East Africa and Madagascar, where people were probably growing bananas by about 500 BC. There were definitely bananas in East Africa by 600 AD. Bananas were growing in China, perhaps brought by Buddhist monks, by 200 AD. Though bananas never really hit it big in China, over time they became a staple food in India and Africa, as well as their homeland of southeast Asia.
When Islamic invaders came to northern India about 800 AD, they liked bananas and sent them back home, so bananas became popular all over the Islamic Empire. By the 900s AD people were eating bananas in Israel and in Egypt. From there bananas quickly reached North Africa, and they reached West Africa by about 1000 AD. From the Islamic sultanates, bananas spread west to Europe with Mediterranean traders from Genoa and Venice. By the 1400s, people were growing bananas on the Mediterranean island of Cyprus, and by the 1500s there were bananas in England.
Learn by doing: eat a banana!
More about modern bananas