The term we often use to refer to the Fertile Crescent is Mesopotamia, a phrase which quite literally means (in the Greek) "between two rivers." This is what has ultimately shaped the region: the presence of the Tigris and Euphrates Rivers. This reflects a major theme of Ancient Civilization—the presence of River Valleys. We could talk about the Indus River in India, the Nile River in Egypt, or the Huang He in China, and see a similar theme: river valleys are central to ancient civilizations and tend to be the oldest locations of human settlement and civilization.
I think the way that you have phrased your question risks misconstruing the relationship between job specialization and civilization—by your wording, you imply that something in the Fertile Crescent allowed the people of Sumer to have job specialization, but ultimately, you should keep in mind that job specialization is actually one of the underlying features which defines civilization altogether. The real question isn't so much why does Sumer have job specialization, but rather, why does job specialization evolve in the context of these River Valleys? Sumer is just an illustration of a much larger and more fundamental theme.
Ultimately, it begins with the Neolithic Revolution (the discovery of agriculture), after which populations became sedentary. What will follow is population growth as settlements grow larger and larger, a population growth which is ultimately supported by agriculture. Here is where we arrive at job specialization: as populations grow larger, at a certain point, it simply becomes both more feasible and ultimately more advantageous for members of a population to start to specialize.
To bring this back to your question about the Fertile Crescent, I would suggest that you keep in mind that this entire evolution is based upon geographical preconditions, and the presence of these rivers created preconditions favorable to ancient agriculture to begin with. There had to be a geography conducive to agriculture to begin with for agrarian civilization to develop, and this is a requirement which the Fertile Crescent certainly fulfilled.