Huge question. The conquests were mostly over by 750. The conquerors were successful because a) they had what in Arabic is called 3aSabiya or roughly "team spirit": they were strongly united by tribal and cultural bonds in a way that city people and residents of the Persian and Byzantine/Roman empires were largely not; b) they offered security in a power vacuum when both empires were at near-collapse states; c) they didn't care about the internal controversies over the nature of Jesus that so preoccupied the areas under Byzantine control; d) they mostly left the city people to their own devices. It should be clearly noted that they did NOT force people to convert to Islam.

When the Abbasids took over in 750, they were usurpers and needed political legitimacy, so they undertook a well-organized program of translating knowledge from all parts of the empire into Arabic and making it available. The idea was a sort of "everyone has something to contribute here", and there was plenty of Hellenistic knowledge, especially medicine and the branch of philosophy dealing with logic and arguments, that the Abbasids valued highly.

As time passed, Islam began to dominate; and Islam is a highly community-oriented faith. Being a Muslim was much like being a Roman citizen centuries before: it guaranteed a certain amount of acceptance in foreign lands and made trade and cultural contact easier.