How, exactly, does this work on cars that have no remote door locks or remote ignition?
If they have Bluetooth this is at least possible in theory. XM radio or other such wireless channels could also be vulnerable, though those are more challenging since they are one-way.
How does this work?
These channels allow a remote wireless connection into one system in the car. While this system is generally isolated from the rest of the car in most vehicles (not in the Model S though) it is in many cases somehow connected indirectly. For example, the Bluetooth system is connected into the audio system, which has some interaction with the car's user interface, which normally runs on a shared Car Area Network that interacts with more critical systems in the car.
Thus, if a vulnerability in the Bluetooth stack (there are often many, some of them well-known) allows the bad guys to create a connection and send some invalid inputs, there might be bugs in the audio or console system that threat these inputs as code (buffer overruns or similar bugs, these are very common in all sorts of computers, though they have started becoming less and less frequent on Windows at least) and that allow the bad guys to run their own code on the console systems. Once there, a hacker can inject streams of data into the shared Car Area Network. Many cars were designed assuming only the legitimate car software has access to this network, so they don't provide any protection once the bad guys are there. Thus, once the bad guy can run arbitrary code on one of the systems connected to the CAN they can do things such as disabling the brakes, opening the throttle or, in more software-based cars even control the steering and more. And at the very least they could open the car remotely and start the ignition.
Tesla being originated from a guy that really gets software I can imagine some significant precautions have been taken, but I would really love to know how much attention has been paid to software security and how formally has this been addressed (there are well-defined and publicly available software security processes that should be followed by anyone developing commercial or critical software).