Current limiting circuit understanding
You have transcribed the circuit from physics.unlv.edu incorrectly.



simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. (a) What you have drawn. (b) The corrected version. (c) The original circuit.
Your R2 should be providing the bias for Q1 and Q2's collector should be connected directly to the base of Q1. This way when the voltage across R1 reaches 0.65 V or so Q2 will start to steal the bias from Q1.
In its modified form of Figure 1c the enable signal provides the bias. The 5 V supply is adequate because Q6's emitter will be at 0.65 V and its base at about 0.65 V above that - so 1.3 V above ground.

From the comments:

OP wants to know about the edge case transition at about 0.65 V and why it is continuous and stable, which this answer does not really address.




simulate this circuit
Figure 2. The equivalent circuit.
I don't think there's much point in discussing the stability of this circuit. It's a pair of PN junctions shorting out VCC to ground. It won't last long.