Why do the drivers turn left and right quickly?

May be...

It cleans the tires, gets the new tires warm, adjust the alignment, and sends a message to the rest of the field that you are about to go fast!!!!

Thats how they clean the cr@p off the tires, & keep the heat in..

It cleans the tires & keeps them hot. Better known as " Scrubbing the Tires"

when they are following the pace car.

It's all about hot rubber. Hot rubber sticks to asphalt better than cold rubber. It's the same reason that drag racers do a burnout - more traction.
Swerving back and forth quickly puts heat in the tire.

They are cleaning off the debris and rubber fragments off the tires. If they don't clean them, then they spin the rear tires on the next restart. This causes them to either get passed by a bunch of cars or a huge wreck.intelli sync lite