Why did the Byzantine Empire fall?
The answer to this depends on whether your instructor wants you to give an immediate cause or an ultimate cause of the fall of the Byzantine Empire.
The immediate cause of the fall of this empire was the Siege of Constantinople in 1453. In that year, Constantinople, the capital of the empire, was conquered by the Ottoman Turks. The Ottomans had been fighting the Byzantines for about 100 years by that time and they had weakened the empire enough that they were finally able to take Constantinople and put an end to the Byzantine Empire.
The ultimate causes of the fall of the empire are the factors that weakened it, making it susceptible to defeat at the hands of the Ottomans. There are many such factors. One of these was the founding and the rise of Islam. Aggressive Muslim leaders attacked the empire, taking important parts such as Egypt from Byzantine control. A second factor was the coming of the Seljuk Turks. They took over much of Asia Minor, cutting of the Byzantines from even more of what had been their empire. A third factor was the Crusades. The Crusades were originally launched partly to help the Byzantines against the Muslims, but the Crusaders (who were Catholic) eventually turned against the Byzantines (who were Orthodox) and took Constantinople in 1204. Although the Byzantines got the city back 60 years later, their empire was weakened by this defeat. Finally, we can say that internal strife caused the empire to fall. There were two civil wars in the empire in the early 1300s, which was both a symptom of internal conflict and a cause of further weakness.
All of these factors helped contribute to the fall of the Byzantine Empire, but the immediate cause of that fall was the Siege of Constantinople, successfully undertaken by the Ottoman Turks in 1453.