Only 3 components are involved in the explanation, the transistor, the base resistor and the rectangle containing the voltage regulator IC.
The 3 emitter resistors are only needed to make sure each power transistor delivers the same current.
It is called CURRENT SHARING .
The IN voltage is about 18v and the voltage OUT is fixed at 12v by the 3-terminal regulator.
The regulator produces the 12v OUT. The transistor does not have any control over the output voltage.
At the moment there is NO LOAD. The transistor is not turned on AT ALL and the output voltage is produced by the regulator. The regulator takes about 10mA to provide this condition.
Now put a LOAD on the output.
This load creates a voltage drop across the 15R and as soon as it is more than 0.6v, the transistor starts to turn ON.
As the load increase (the resistance of the load decreases) by say adding more globes, the current increases and the voltage-drop across R increases and this turns the transistor ON more.
The 3-terminal regulator (in the rectangle) delivers almost no current to the load during any of these conditions (only about 50mA !! ). It is the transistor that delivers all the current.
The current PASSES through the transistor and that’s why it is called a PASS TRANSISTOR.
The purpose of the 15R resistor is SOLELY to allow the transistor(s) to turn ON.
As the load takes more current, the transistor will be "asked" for this current, but it does not deliver.
So, the regulator delivers the extra current and this current flows through the emitter-base junction. The 15R is added so that most of the current flows through the resistor, or at least some of the current flows through the resistor.
The extra current requested by the regulator flows through the emitter-base junction of the PNP transistor and turns it ON more. This allows the transistor(s) to pass the current from the bridge to the output.
I don't know why the 15R is rated at 10 watts. The voltage across this resistor cannot rise to more than 1v and this means the resistor can only pass 1/15 amp = 60mA !!
Obviously the circuit has never been tested.