Step 1: Dissemble Watch
Step 2: Route Switch Wires
When there are 0V across RED-BLK, the transistor is non-conducting.
When there are > 0.6V across RED-BLK, the transistor conducts.
When there are < 0.6V across RED-BLK, the diode protects the transistor from reverse breakdown.
The 10K resistor limits current into the base.
Transistor turn-off speed isn't critical in this case, but if it was, I'd recommend adding a 1K resistor in front of the 10K resistor and a 10-100nF capacitor in parallel with the 10K resistor. This would help to quickly add to, and remove base charge to the transistor.
The flash card with the transistor on it shows how to figure out the pinout of an unknown NPN transistor. Using a multimeter in diode test mode, you'll find that the base-emitter junction has a higher potential than the base-collector junction.