In typical designs, the car continually transmits a low-frequency (e.g., 135 kHz) radio signal to wake up any wireless keys within range. When a key receives this signal, it replies with a VHF (e.g., 315 MHz) signal, and the car unlocks or starts when a door is opened or the start button is pressed.
The reply signal, at least, is uniquely coded to the car. The attack is to extend the range of the LF wake-up signal, causing a key stored away from the car to transmit a valid reply.
In some models, besides the transponder described above, the key also has a passive RFID tag, which works with a reader in the car to allow starting even if the battery in the key is dead.
(The article is wrong about the broadcasts, by the way; if the key transmitted continually, its battery wouldn’t last long.)