The Great Awakening led to (or helped lead to) egalitarianism because it conveyed a message that implied that individuals' relationships with God were more important than doctrine and the teachings of church leaders.
The preachers of the Great Awakening emphasized the need for people to have emotional relationships with God. They argued that religion was a matter of the heart and of emotions, not of erudition and book learning. By emphasizing this, these preachers weakened the position of church leaders. They told people, in essence, that the leaders were not necessary and that people could find God on their own through emotion. They did not need to be learned in order to understand what God wanted.
By preaching such a message, the Great Awakening helped to instill the idea that the common people were capable of running their own affairs in something as important as religion. If this were so, there was no need for hierarchy in society either. This was a message that asserted the equality of all people, thus leading to egalitarianism.