What are the motives behind Arab expansion?

The Arabs could expand and they perceived that doing this was their duty as pious Muslims.

Historically, there was the "Right to Conquest" which was a pervasive idea in political thought. The idea was that it was natural for any country or state to grow and control more territory as it grew stronger. This allowed weaker states to dissolve in place of ones that were better run, a bureaucratic version of "survival of the fittest". The "Right to Conquest" prevailed as the dominant theory of nation-building until the mid-1800s when people started bringing up the concept of self-sovereignty and ethnic nationalism, which held the idea that people should govern themselves even if they are not the most powerful in the world. This view of state sovereignty has become dominant today and the Right to Conquest is seen as incorrect.

Since the Arab Caliphates existed well within the Right to Conquest Period, the Arabs did not have to assert a reason to conquer neighboring territories in Southwest Asia. It was their natural prerogative.