How did the Sumerians develop successful agriculture?

By using the waters of the two big rivers the Tigris and Euphrates in sophisticated ways. They canalized its waters, so that during the dry periods they could use it to irrigate their fields. The yearly inundations also left a big part of the land covered with a very fertile mud. These two natural and created factors made the "fertile crescent" a very good place to create one the first civilizations on earth.