Why does it work? One needs to understand static gas temperature, total gas temperature and propulsion if a proper physical picture of the effect is to be constructed. This is an article by me with Prof. Straatman that explains the fundamental law of rotational cooling, it assumes a sophomore-level math and physics:
Polihronov, J.. Straatman, A.Thermodynamics of angular propulsion in fluids, Phys Rev Lett 109 054504 2012
Also, see this web site, I am putting together an easy-to-read explanation of the vortex tube effect
In more detail -
Consider the concept of vortex flow "discretization": let's simplify the vortex flow by introducing a simple flow system, which still exhibits the physics of temperature separation. The simple flow system comprises a rotating adiabatic duct and a tank of compressed gas attached to the inlet of the duct. The outlet of the duct is at $r=0$, while the inlet is at $r=R$, the point $0$ is the rotation center.
Set the system into uniform rotation. Let the linear speed at the inlet is $c= \omega R$. Then, in the stationary frame of reference, the total temperature of the gas at the inlet (at periphery) is $T=T_0 + c^2/2c_p$, where $T_0$ is the static temperature of the gas in the tank. From rothalpy conservation we get the total temperature at the outlet (at center) to be $T=T_0 -c^2/2c_p$, $c_p$ is the isobaric heat capacity of the gas. Thus, the total temperature separation is $\Delta T=c^2/c_p$. What happens with the static temperature $T_s$? At inlet (at periphery), $T_s=T_0$; at outlet, it is $T_0-c^2/2c_p$. Thus, the static temperature separation is $\Delta T_s=c^2/2c_p$.
Temperature separation is observed in a rectilinearly moving system as well. Consider an elemental system, comprising an adiabatic duct and a tank of compressed gas attached to the leading end of the duct. Set the system with uniform linear velocity $c$. Let gas leave the system with velocity $0$ in the stationary frame of reference, the gas comes out the trailing end of the duct.
The temperature separations $\Delta T$ and $\Delta T_s$ are exactly the same as in the rotation case; only now conservation of enthalpy needs to be applied to solve for the temperatures.
When it comes to the vortex tube effect, this analysis is a very good place to start.
I hope this helped!