The fix is actually really simple. Technicians just have to fit a flow transformer onto the mass air sensor housing assembly. In English, they have to stick a little plastic tube outfitted with a grate (the "flow transformer") over the thing that regulates the car's fuel injection (the "mass air sensor") This sensor does its job by measuring how much air is moving through the intake manifold and adjusts the amount of fuel being injected into the cylinders accordingly.
The grate will calm and stabilize the airflow before it hits the sensor, resulting in more accurate readings. Technicians will also have to update the car's firmware though, according to the video, the entire installation and update process shouldn't take more than an hour. As for their 3-liter gas engines, VW says they can get by with a software update alone.