It was settled because it was an extremely fertile place to grow crops. The word "Mesopotamia" literally means "between the rivers", and refers to the stretch of land between the Tigris and Euphrates rivers in modern-day Iraq. The earliest civilizations, places of concentrated urbanism, were in very fertile areas where water was plentiful and crops could be easily grown, sometimes multiple harvests a year. Other areas where this is the case include Egypt (Nile river) and the Indus River Valley in India.

Why does civilization crop up in these areas? Mostly because the process of urbanism is about specialization. People settle together in one place for protection. Once they begin farming, this is necessary because it's easier to raid a single home in the middle of nowhere than it is to raid a walled city. Once farming is established, you are now producing more food than is needed by the number of farmers. This means people can exclusively devote time and effort into specialized tasks, such as metallurgy, pottery, bureaucracy, crafts, etc. At this point, you have what is known as "craft specialization". This is all made possible by the surplus of foodstuffs produced by farming specialists, and the easiest places to do that are in areas that are well-watered, such as the aforementioned river valleys.

http://www.regentsprep.org/regents/global/themes/change/neo.cfm

https://archive.org/stream/CivilizationAndUrbanismRiseOf/c#page/n0/mode/2up

As for your second question, no, the Mesopotamians never "moved over" to the Mediterranean. The indigenous people groups of the Mediterranean coast followed the same general process of learning how to farm and develop urbanism and specialization and arose into their own kingdoms and polities (Aram, the Hittites, the Canaanite city-states, ancient Israel, the Phoenicians). They just took longer to get fully settled because it is harder to farm in a rain-fueled agricultural system than it is in a river-basin flood agricultural system. It took time for the technology to allow them to settle. There were still empires in Mesopotamia when the Phoenicians were beginning their domination of the trade ports of the Mediterranean (King Nebuchadnezzar of Babylon in 586 BC attacked them and conquered them). There were even still empires in Mesopotamia when Alexander the Great went through and conquered them (battling King Darius of Persia).

http://en.wikipedia.org/wiki/History_of_the_Middle_East

The Phoenicians are a particularly interesting case. They began as Canaanites, but when the Sea Peoples and Israelites took over and pushed them out of Canaan, they were relegated to a tiny area on the coast, much of which was mountains. Not a very fertile land at all. This meant that they needed to base their economy on maritime trade, and they ended up building a trade-fueled empire spanning the entire Mediterranean, founding colonies such as Carthage and reaching as far as Spain.

http://en.wikipedia.org/wiki/Phoenicia

EDIT: sources