A two-wheeled motorcycle allows the rider to turn around corners by leaning and making use of the gyroscopic effect. The further the lean, the turn gets tighter but more risky due to instability and loss of traction on the wheels.
Unless you are a competitive motorcycle racer, the risk of falling during a bad turn could be higher due to various conditions, including slippery roads (after a rain), dust, sand, etc. Piaggio designed the MP3 to solve that problem by introducing 2 wheels to the front to improve the stability and traction. However, the most critical part of the invention is the patented parallelogram linkage that allows the two front wheels move as if its a single front wheel.
In short, the design increases the safety and stability of turning without sacrificing much of the turning maneuverability.
Read also article at NY Times