The regulator adjusts its cathode voltage to maintain a constant voltage at the reference input. It's like a reference voltage plus an op-amp with a transistor.
The external transistor will control current depending on the base voltage. So by controlling voltage at the emitter (where the REF input is connected), the collector current (which is about 99.5% of the emitter current) is effectively controlled to within a fraction of 1%, provided you don't run out of compliance voltage. The collector current will be approximately Vref/R2, or about 2.495V/R2
There are slight errors because the transistor gain is not infinite and the REF input draws a couple microamperes, but it's a pretty good current sink. The collector can't quite get to Vref, so the minimum collector voltage is about 2.6V, meaning the voltage across a load to +5 can't be more than about 2.4V/I.
Note that the collector current is what is controlled to be constant. The circuit block is a constant current sink. If you measure the collector current and open up R4 then the collector current should not change (unless you exceed the permissible range of voltages at the collector).
In this case, the current is 2.495V/220\$\Omega\$ = 11.34mA. The maximum load resistance (collector to +5) is about 2.4V/11.34mA = 211\$\Omega\$. When you exceed that resistance you no longer have a constant current sink. So your circuit with 1K is not a useful working example.
The emitter current as a consequence is also constant but that's pretty useless as it changes with R2 so it's more of a set voltage across a resistor.
Here is a quick LTspice simulation with a 100 ohm resistor as a load:
Ic(Q1): 0.0113147 device_current
Reduce the 100 ohm resistor (R3) to 0 ohms and we have:
Ic(Q1): 0.0113152 device_current
So it's a pretty good current sink. Output resistance (ideally it would be infinite) is about 2M\$\Omega\$.
Similarly, if I increase the supply voltage to 10V with R3 constant at 100 ohms, I get:
So about +0.06% per volt line regulation. That could be improved by increasing R2. With R2 = 1K the line regulation is improved to +0.01%/volt and load regulation is slightly better too.