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When the fans and fins press you down you have more traction, so you can turn harder and faster without rolling or sliding

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So, the Tesla demonstration is just to show that they can make the car more "stick" to the road and then go faster. Do I understand?

That's the practical purpose, faster cornering, faster takeoff. It's been tried before as well. For example a version of this car from 1978 Formula 1:

https://en.wikipedia.org/wiki/Brabham_BT46

"The "B" variant of the car, also known as the "fan car", was introduced at the 1978 Swedish Grand Prix as a counter to the dominant ground effect Lotus 79. The BT46B generated an immense amount of downforce utilizing a fan, claimed to be for increased cooling, but which also extracted air from beneath the car."

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