If you have an oscilloscope, you can put the scope probe on the antenna of the transmitter with the ground clip of the probe to the battery negative. You could try coiling a length of wire around the antenna and putting the probe across the resulting coil to get a viewable signal. You'll likely see pulses of RF as you manipulate the controls on the transmitter. The repeating pattern of the pulses is different for the various control positions.
Edit: You can probe on the toy car circuit board to see the pulses too, but it's going to be a bit of hunting and pecking without the schematic, and you may still not be sure the pulses you are seeing are those actually received by the antenna.