Most likely, the transistor is being used as the switch.
Transistors have three terminals; 1) Base, 2) Collector, and 3) emitter. In this application, I'm going to guess that the power is tied to the collector and that when the transistor is forward biased, voltage and amperage flows from collector to emitter thereby providing power to the motor to run.
My suggestion is that you take the circuit board to an electronics repair shop. If my suspicions are correct about the electrical flow, just about any radio or TV shop should be able to make a reliable repair.
The thing you should be worried about is WHY the circuit became overloaded. Transistors are rated on the maximum amount of amperage capable of flowing through the transistor from collector to emitter. If the original transistor just wasn't able to withstand the normal amperage during the life of the vehicle, no problem; just replace the transistor with one able to withstand a larger amperage flow.
So, the first thing would be to measure the amperage flowing in the circuit to make a determination if the original transistor was of the correct size to handle the current. Take the measured amperage information with you when you transport the circuit board to your chosen repair shop. Make sure you take note of the "start-up" amperage and not just the "running" amperage.
Transistors are manufactured in NPN and PNP configurations. NPN is the most common type. You must replace the defective part with the same type or other steps must be taken to ensure that the resistors which bias the transistor are of the correct size and are placed to cause the transistor to forward bias properly.
Transistors have the ability to use a DC voltage to carry an AC signal. Most likely, the circuit board toes not generate an AC signal, but that cannot be ruled out until it is looked at by a qualified technician. Another component that is likely present on the circuit board is either a 555 or 556 solid state timer used to control the amount of time the transistor is switched on..
Unless you have the correct level of training to replace the transistor yourself, I would strongly suggest that you find a qualified technician. Transistors may be biased using several different techniques depending on the application. I suspect that the transistor in this particular application is "based biased". If you do not understand these terms as they are applied to transistors, do not attempt to make the repair yourself!
A discrete electronic tutorial is beyond the scope of this medium. Please do not ask for further explanations for any of the aforementioned terms. The summary above is a two year culmination of both analog and solid state electronics training including many mathematical formulas needed to properly bias transistors in single and multiple stage amplifiers.