They are accused of this:
-Selling millions of cars that are not certified. VW does have some certificates, but they are valid only for (non-existing) cars without the defeat devices. Devices that work like this have to be declared. There are acceptable uses for example when handling emergencies or when starting the car, but cheating on emissions testing would presumably not have been an acceptable use. So VW didn't mention them in the application and thus the certficiates they do own are not valid for the cars they sold.
-Having installed defeat devices, which is illegal.