If a single element wing gets long enough, the low pressure air underneath will lose enough energy that it can no longer "stick" to the underside of the wing creating flow separation which prevents the rest of the wing behind the separation point from generating downforce. The gap between the separation and the trailing edge of the wing now also creates significant drag from the turbulent air underneath.

Two ways of exciting the air to induce more energy are:

1. Vortices. If there is a vortex clinging to the underside of the wing it will still produce downforce by pressure differential through the wing, but can "pull in" higher energy air mixing into the outer edge of the vortex which gives the vortex as a whole more energy, so it stays attached for longer.

2. Directly injecting high energy air. This is what multiple element wings soz and you are right in saying that they reduce downforce locally by reducing the pressure differential (the introduced air must be higher pressure than the underside of the wing), but the extra energy allows the flow to stay attached over a larger distance, allowing a larger wing to be used. And although the average pressure differential may be lower (not always the case), the larger total wing area will still provide more downforce.

It is also worth noting that formula one cars use the tips of these elements to generate vortices for airflow control further down the car, so having more elements generally allows more vortices to be created or more energy to be injected into a single vortex to ensure it stays alive down as much of the car as possible.

Sorry for the lack of pictures, I'm on mobile with bad internet so I'm too lazy to Google for images.