Looking at the schematic, it's easy to see why the Darlington Transistor (MPS-A13) fails. When an inductive load (real, motor, etc) is switched on/off by a transistor, at the moment the transistor turns the motor off, the inductive load wants the current to be contents and expends its energy trying to do so by creating a spike of voltage. If the voltage is higher than the breakdown rating of the transistor, it's time for a new transistor.
What is normally done is a diode is put in reverse across the motor or relay to limit the voltage to .7V and not kill the transistor.
If you look at the KLine relay board, my guess is the diode on the back side is across the relay coil to protect the transistor.
A simple analysis of how the circuit works is that when AC with an average value of 0 V is on the track the Darlington transistor is in the off state. When the whistle button is pushed, the average value of the AC increases and the Darlington transistor conducts and allows current to flow through the motor.
The motor is obviously a DC motor due to the diode and filter capacitor.