In this kind of regulation...
This circuit is not really a regulator as there is no true feedback control loop.
I repeat the image from the original post for discussion:
The idea is that both Vz and Vbe are comparatively independent of load current and, to a lesser part, of input voltage:
The differential (Vin-Vz) drives a current Is through Rs: Is = (Vin-Vz)/Rs. This current is split into one part Iz going through the zener diode and anther part Ib going into the base of Q1.
The part Iz will create a stable zener voltage Vz.. A change in Iz will result in a small change in Vz:
(
sourced from here with modifications).
Therefore for practical purposes Vz = constant.
Equally the base emitter voltage Vbe of an NPN transistor is, above a certain minimum current only weakly dependent from emitter current (emitter current being the sum of base current
and collector current!).
(
sourced from here with modifications).
Therefore, again, for practical purposes Vbe = constant.
In total, again for practical purposes, Vout = Vz + Vbe = constant.
Note: practical purposes here means within and as long as the resulting tolerances of Vout are acceptable in the application.
This is the most basic explanation I can think of for this kind of voltage "regulator". A more detailed consideration by a network analysis using the transistor and diode equations is left to you. You've been given a few links to study.