A diffuser allows the air traveling underneath the car a place to expand and decelerate back to road speed as well as providing wake infill. As the air enters towards the front of the car it accelerates and reduces pressure. There is a second suction peak at the transition of the flat bottom and diffuser. The diffuser then eases this "high velocity" air back to normal velocity and also helps fill in the area behind the race car making the whole underbody a more efficient downforce producing device by reducing drag and increasing downforce.
Understand that a true flat bottomed car (one without a diffuser) will produce downforce in and of itself when run in rake. Essentially the entire flat bottom becomes one large diffuser. It too has two suction peaks, one upon entrance, the second at the trailing edge of the flat undertray. A diffuser acts to enhance this underside suction, it acts like a pump, encouraging better flow under the car.
One thing to note is that the rear wing interacts with the diffuser "driving" it. The proximity of the low pressure side of the rear wing encourages better flow through for the underbody.