What is the main reason for the growing independence of medieval urban areas?

The revival of commerce.

The feudal structure of medieval life was based around landowners and the agricultural produce of those who worked on their lands.  It was a system built for stability, not growth.  As towns were established, they sought autonomy from the local lords and became places where merchant guilds and businesses thrived.  They came to operate essentially independent from the normal feudal structure of the otherwise agriculturally-based economy.