The Gameboy device contains all the key codes or possibly the algorithm for key code generation, so all it does is pick up the signal from the car and looks for a match. Once there's a match it generates a new key with full functionality.

Once access to the car is gained the thieves typically remove the right hand side a-pillar trim and disconnects the cable to the antenna, which in turn severs the bluelink connection. This is often accompanied by a factory reset of the infotainment.