Begin by assuming that the 12V dc power supply is able to provide a lot of current at 12V while suffering no voltage drop. In other words, the power supply has a very low source resistance. This assumption helps establish how to analyze the circuit using ohm's law. In ohms law there are three variables, Voltage, Current and Resistance. If you know two of these, you can calculate the third for any circuit path. And we know that the 12V terminal always has 12V no matter how much current is drawn from it.
Next, we consider how much current will flow through the resistor. One side of the resistor is connected to the 12V supply, so that is fixed. On the other side of the resistor is the zener diode connected to ground, and the base of the transistor. At the moment, there is nothing connected to the emitter of the transistor, and we know that it is a NPN transistor, so we can say that no base current will flow since there is no path for it to flow out of the emitter. So all of the current that flows through the resistor will also flow through the zener diode. Now, look at the zener diode current vs voltage curve. It tells us that as the current starts to flow through the resistor and then through the diode the voltage across the diode will flatten out and remain at the "zener" voltage once the current is high enough. The current will then come to rest at a specific value which is easily calculated. Since the voltage across the diode is now known, we calculate the voltage across the resistor to be 12V minus the diode voltage, or 2.5 volts. So now we can calculate the current through the resistor to be 2.5 divided by 1Kohms, or 2.5 mA. This is also the current that flows through the zener diode. The voltage at the junction of resistor and zener diode is thus "stabilized" and remains at one value even if, for some reason, the current amount flowing through the diode happens to change. But how could this happen?
So far, we do not have anything connected to the transistor's emitter. To make some use of this circuit, you attach a load to the emitter, a load with some value of resistance from emitter to ground. Let's start by attaching a load of, say, 800 ohms (just to pick a value out of the blue).
At the instant that we apply power to the circuit with the load attached, there is now a place for emitter current to flow out of the transistor. As the voltage ramps up, the transistor base voltage increases, and thanks to the action of the transistor, its emitter voltage also ramps up, at roughly 0.7 volts lower than the base voltage. The current that flows into the base stabilizes at the amount required for the emitter to achieve 8.8 volts across 800 ohms, divided by the beta of the transistor (the DC current gain). So we can calculate the amount of base current that flows, or rather, we can estimate it pretty closely anyway. It would be 0.11 mA.
This calculation is based on the assumption that the base voltage remains steady. This happens to be true even though now we are taking some base current. The reason for this is that even though it would seem like we should be taking a bit more current through the resistor now, in fact we are not. What is happening is that we are stealing a tiny bit of current from the diode branch and feeding it into the base instead. Since the diode voltage drop remains stable at 9.5 volts for such a small reduction in current through it, the diode voltage does indeed remain steady.
Now, imagine that we change the load resistance from 800 ohms to some other amount. What happens is that the transistor then pulls more or less current into its base, and is stealing a little bit more or a little bit less current from the zener diode. But the zener has this handy characteristic that the voltage across it doesn't change when this happens. And since the emitter of the transistor remains 0.7 volts lower in voltage than the base regardless of small changes in current low through it, the transistor emitter voltage also remains steady.
That's how this circuit "regulates" the voltage across the load.