I know it started at the same time in the Indus and China. Buy why on earth in the desserts of the Middle East? Why not Italy, or Egypt, both of which had great farmland and were known for it during the time of Rome.
Civilization started almost simultaneously along great rivers - either Eufrat and Tigris, or the Nile, or Indus. The reason for it was that people needed to cooperate on irrigation projects - the soil along the rivers was very fertile.
The deserts were actually a key factor in the rise of Mesopotamian civilizations. In areas of greater natural fertility, humans could survive in small groups, living off the land. In a hostile environment, people had an incentive to group together to overcome the challenges presented by nature. The building of irrigation systems to harness the major rivers and flood plains required cooperation between large groups of people. Additionally, the concentration of arable land around the rivers forced people into closer contact.
As groups of humans grew in size, methods of communication and organisation became neccessary. Most importantly, this led to the development of writing systems (some of the earliest written evidence from the region is economic in nature - records of transaction and so forth).
I took a year's worth of classes on Mesopotamian history, which by no means makes me an expert. This is just a summary of what I learnt in lectures.