Anthropology and historic excavations provide some clues to answer your question. For the purpose of the answer, we will define more complex societies as societies that formed recognizable governing structures. Let's begin at the beginning!
The earliest known societies were nomadic tribes crisscrossing the African and Euro-Asian continents. These tribal communities shared a common mode of communication, banded together on hunting expeditions, and in general exhibited a basic organization structure allowing the tribe to survive the harsh conditions they encountered. The constant search for food was the main objective for the tribe to remain together, and until the tribe located in an area plentiful of resources (food, water, and shelter), there was little need for civil authority as we think of it in our time.
The first organizational structure was tribal, probably led by one leader or a group formed into a council. For the group to evolve beyond a nomadic tribe with a small cadre of leaders, three events would occur, resulting in the need for more complex organization. The first evolution is for the tribe to become stationary and begin to settle in place of nomadic excursions. For a tribe to remain in an area, there has to be more than sufficient resources to support the tribe. However, while resources are essential to the survival of the tribe, if the tribe does not recognize the abundance of resources or does not have the technology to exploit the resources, the tribe will not survive. An example of how quickly nomads became settlers can be found in the ancient civilizations spanning the Nile River and the Nile River Valley. Once the nomadic tribes trade their hunting weapons for farm implements, civilization as we think of it explodes. The Fertile Crescent in Nile River region provides some of the best examples of the evolution from nomadic to civil governing societies.
One of the earliest known civilizations is the Sumer civilization—better known as the Sumerians in Mesopotamia. The Sumerians represent the second stage in the development of more complex society. The Sumerians were terrific at agribusiness and produced remarkable amounts of agricultural products in surplus, creating a dynamic trade with surrounding cultures. The second evolution is production capacity to exceed basic need and the resulting creation of a dynamic trade. Trade and a functional economy relies on civil stability, productive technology, and common trade language; all three of which flourished in Sumerian society. Complex societies have viable economies as the result of established trading norms such as currency rates, language, mathematical ability, and regulations.
Two of the main conditions for simple societies to become more complex are stability and technological advancement, both of which create a dynamic economy.