Car kits were allowed to go up to 25W transmit power, but typically operated at 1W or so most of the time, as they had a good antenna. The control uses the microcontroller on the board to do the transmit receive switching in synchrony with the handset ( they communicate via a 1-2 wire interface on the link, so that the phone knows to turn off it's antenna and to accept audio from the car kit and send audio there, along with then doing the car specific functions like mute the radio, respond to the commands from the car kit for volume and such) so that the GSM/CDMA frames are properly handled. The switching is done partly by the large sealed module, which has a set of PIN diodes to isolate the transmit and receive filters ( which have further pin diodes to select the various GSM/CDMA channels in the 800-900MHz spectrum allocated for GSM/CDMA) and switch rapidly between the receive pass through ( most of the time) for the very short transmit pulse in the allocated time slot for the handset.
The Exar chipset is likely the audio hands free kit, having AGC, echo cancellation, bandpass filtering and muting to get the audio from the microphone mounted near the driver and remove the speaker audio ( mostly) and send it up the cable to the handset, with the handset audio coming down another wire to it for the speaker ( it has an amplifier of around 5W there, with a relay to disconnect one car speaker to use for audio, or uses a mute line on the radio to mute it and use a separate speaker) for the other party audio. There is also going to be some circuitry to charge the phone via the cradle, and to sense car state, turning off the charge when the ignition is off unless the phone battery status is low, and general power supply for the parts.
Sadly not much use any more, as the 800MHz band is only rarely used these days, most phones using higher bands to get more capacity. Only a few small carriers use it, as the equipment for it, though very expensive new, is cheap as used but working stuff from other operators upgrading systems to increase capacity. Many phones no longer have a 800MHZ Rf section as well. Only bonus with 800MHz is that the cells can be very large, which is an advantage in rural areas where most of the old systems are, as it covers a much larger area but must have a lower customer density to avoid overloading the base station. You can easily have a 60km range from such a cell with little dropout, though at 60km the handset will be transmitting at the full 5W of power. Most customers with that use have fixed high gain directed antennas pointed at the tower and a very local passive dipole joined by ultra low loss coax, using the phone as a semi fixed base station.