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 and co-authors that explains the fundamental law of rotational cooling (also known as Euler's turbine equation), it assumes a sophomore-level math and physics:
Polihronov, J. et al, 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.
To sum up, the vortex tube phenomenon is an example of Euler's turbine equation at work.
When it comes to the vortex tube effect, this analysis is a very good place to start.
I hope this helped!