There is solid evidence to suggest that the Arab conquests following the death of Muhammad served as a continuation of his work. During Muhammad's lifetime, Islamic leaders embarked on a program of consolidating power and spreading the influence of Islam. "Islam," meaning "submission," expresses the need to submit to the will of Allah. In the time of Muhammad, and in later centuries, this was interpreted as extending this opportunity to submit to other people. It was perceived as the fulfillment of Islam's destiny to pursue conquest in the name of reclaiming the world in the name of Islam.
One could also argue that the process of conquest was a natural continuation of Muhammad's work in a political and economic sense. Politically, it served to create some advantage to consolidate authority. Economically, much of the territory acquired through conquest granted the holders of that land with control over the trade of that area.
Ultimately, while political and economic factors could be said to speak to the importance of the Arab conquests of the seventh and eighth centuries, the ideological component of Islam itself proves even more revealing as a possible answer.