How did the development of a surplus in primitive agricultural society allow for occupational specialization and possible social stratification?

A vert short answer is: people had time.

Before this, people were focused on not starving and everyone's attention was turned to gathering food. Noone could afford doing anything else than gathering food for themselves (and maybe their offspring) because then they would starve. But, with the surplus of food, some people could focus on innovations and those innovations (such as writing) gave them a higher status, which ultimately led to a stratification.