Egyptian Arabic is dominant in a similar way to American English, i.e. Egypt had the largest movie industry and population, and everyone else could understand Egyptians while Egyptians had no need to understand other dialects.
What is interesting is the impact of satellite television. These new stations have global audiences and content that Arabs want to watch as opposed to their national censored channels. But their solution to pan-Arab broadcasting is classical Arabic, not Egyptian Arabic, and so there is a resurgence of the use of classical Arabic outside of courts and classrooms.