Describe why people settled in the Balkan Peninsula by the A. D. 400s and
500s.

People especially Germanic tribes started settling during the Great Migrations, that was started by the Huns. They were practically obliged to move from the region of Eastern Europe to the region of Eastern and Western Empire. Additionally, one important reason for their removal was the lack of land that could produce them enough food.

Explanation:

Germanic tribes lived across the border of the Danube river, and were divided into many tribes - Ostrogoths, Visigoths, Vandals... They were often mercenaries in Roman Empire but still lived outside the border of it. But, when the Huns came to Europe they started settling in the Balkan Peninsula. They were ravaging the territories, conquering cities, even killed a Byzantine emperor in the battle.