It may be easier to understand this circuit by first considering it without the transistor. In that scenario, all the current to the output must flow through the 68R resistor and the LM2931. If a circuit connected to the output is drawing 1 mA, then you will have 68mV dropped across the resistor and the rest of the remaining voltage between the input and output dropped across the LM2931. For example, with a 10V input, you will have (10 - 5 - 0.068) = 4.932 volts dropped across the LM2931.
If the circuit connected to the output draws 10 mA, then you will have 680mV dropped across the resistor and, again with a 10V input, 4.32 volts dropped across the LM2931.
If the circuit connected to the output tries to draw 100mA, then you will 6.8v dropped across the resistor, leaving only 3.2v for both the LM2931 and the output. At this point, you've overloaded the circuit.
If you repeat this with the transistor in place, what happens?
For a 1mA load, you have 0.068v dropped across the 68R resistor, and therefore 0.068v across the base-emitter junction on the transistor. The transistor is off and no current flows through it. The LM2931 is delivering all the current to the load.
For a 10mA load, you have 0.68v dropped across the 68R resistor, and therefore 0.68v across the base-emitter junction on the transistor. At this point, the transistor is starting to "turn on" (for a very simplistic transistor analysis). The LM2931 is delivering most of the current to the load, but a little bit is starting to flow through the transistor in parallel with the LM2931. Note that the LM2931 is controlling the output voltage.
As the load increases above 10mA, the LM2931 will deliver more current. The current delivered by the LM2931 will mostly flow through the 68R but a little bit will flow through the base of the transistor. As the voltage across the 68R resistor increases above 0.7v, the transistor will turn on. More current will flow through its base and more current will flow through its collector, bypassing the LM2931 and being delivered directly into the load.
At this point, the transistor is acting as a current multiplier. The more current the load tries to draw, the LM2931 will deliver a bit part of the load, increasing the current across the 68R resistor, and therefore increasing the base-emitter voltage on the transistor. This turns the transistor on "harder" and therefore the transistor also supplies more current to the load.
For any load that is significantly above the 10mA threshold that starts to turn the transistor on, the LM2931 will deliver approximately 1/(1+hfe) of the transistor and the transistor will deliver the rest of the current.
[To all transistor experts out there, the above is very deliberately simplified to (hopefully) make the general behaviour more understandable. I know that it's not in any way accurate.]