What exactly is reactive power, in concrete terms?
The imaginary part is energy that is bouncing back and forth between reactive components without being dissipated. Essentially, you can think of it as energy that gets stuck in the wiring due to the existence of capacitance and inductance.

Now to your actual question:


  I am trying to understand how the mathematical expression Imaginary(s) is what you just described vaguely. 


Complex power = V^2 / impedance.  The impedance is a complex value, with the real part being resistance.  The imaginary part is reactance.  If you take the imaginary part of complex power, you get V^2 / imag(impedance) = V^2 / reactance.  So the imaginary power is just the effect of voltage on reactance (the imaginary part of impedance).