VW's cheating is in essence the cars have two modes to run the engine; compliance mode and performance mode, and they run in compliance mode when the car starts up until the steering wheel is turned (more or less).
In compliance mode, the engine control follows the rules to meet the emissions test standards. In performance mode, operating temperature is allowed to increase, which increases performance, increases fuel efficiency, idles better, etc, but increased operating temperature leads to more NOx emissions.
Additional, for models with diesel emissions fluid, performance mode injected much less DEF than compliance mode; DEF reduces operating temperature as well as directly reacting withe NOx. This reduced use of DEF allowed a smaller DEF tank to be used; regulations require passenger car DEF tanks to have enough capacity for normal use within the regular service interval; if the vehicle was operating in compliance mode the whole time, you'd need to fill the DEF tank between oil changes (or have a larger tank, which needs to fit somewhere).
In the end, the big tradeoff is fuel efficiency (and therefore CO2 emissions) vs NOx emissions; which is a fine tradeoff to consider, but you can't give drivers what they want and regulators what they want without cheating.
Before the shift there was an influx of tiny cars with tiny diesel engines which had emissions <= 99grams CO2 / km.
Now it's NOx all around. In fact, The Netherlands has limited daytime speeds on highways to 100km/h vs 120/130km/h just for this purpose.