One of the realities of pressure is that it acts in all directions. Thus the relatively high stagnation pressure at the front of a moving vehicle pushes back on the vehicle (creating drag). Now if we stick a splitter out on the bottom of the spoiler, then the stagnation pressure will also push down on the top of the splitter. But what is the pressure on the underside of the splitter? If it is the same as the pressure on top then there will be no force on the splitter. So let's examine the pressure underneath the splitter.
Since the splitter is close to the tarmac, it creates a restriction to air flowing underneath it (a mini-Venturi if you will). Just as water in a river speeds up when the river narrows, the air that is piled up in front of the spoiler must speed up if it wants to squeeze underneath the splitter. And as we know from the Bernoulli equation, when an airstream speeds up, its pressure goes down. Thus the region between the splitter and the pavement is an area of low pressure. To recap; high pressure exists on top of the splitter, and low pressure exists beneath it. This adds up to a downward force on the splitter. And this downward force is directly proportional to the surface area of the splitter. So, up to a point, the bigger the splitter is, the more downforce it can create.
Two photos of splitter implementations are shown below. Notice the enormous size of the splitter on the Panoz Le Mans car. And on the Ford Mondeo BTCC car we can clearly see the very narrow "venturi" section under the splitter. This speeds up the air flow and lowers the local pressure.
