There are a number of factors that could help to account for the rise of Islam in the time during and soon after the life of Muhammad.
First, we have to assume that the content of the religion itself had something to do with its spread. The theology of the religion and its emphasis on monotheism and community must have been appealing to people in the areas that it came to dominate.
Second, the location of Saudi Arabia was important. This was a crossroads for trade. Because of that, the ideas of Islam could spread with traders and come to appeal to those in neighboring areas.
Finally, the military prowess of the Muslim armies had something to do with the spread of the faith. The armies were able to conquer far and wide and that enabled them to spread their religion to far off areas.