Now I understand that separating two attracting charges requires
energy, which is stored as the potential energy of the "capacitor
plates".
The energy is stored in the electric field between the capacitor plates.
But when assigned to a non material object, like energy in a "region"
I can't understand what it means at all, nor can I form any intuition
regarding this.
It isn't any "region". It is a region where a field exists. For the capacitor, the energy is stored in the electric field of the region. Similarly, in the case of gravity, the energy is stored in the gravitational field of the region.
Could you tell me what "Energy stored in an electric field" is?What is
its practical interpretation?
One definition of energy is it is "the capacity for doing work". An electric field (and gravitational field) has the capacity to do work.
Suppose you have an charged air capacitor, i.e., a capacitor having two plates separated by an air gap. Energy is stored in the electric field in the gap. As proof, if you could place an electron somewhere in the gap it would experience a force due to the electric field causing it to accelerate towards the positively charged plate. That electric force has done work on the electron giving it kinetic energy.
Again, suppose we have a capacitor that was charged by a battery and then the battery removed. There remains a voltage across the charged battery and energy stored in the electric field between the plates. If you now connect say a resistor across the capacitor terminals a current will flow causing heat dissipation in the resistor. Where did the energy come from to create the current and heat? From the electric field between the capacitor plates. Eventually the current will stop when the voltage falls to zero and all the energy stored in the electric field of the capacitor has been used. That initial stored energy was
$$E=\frac{CV^2}{2}$$
Where $V$ = the initial voltage of the fully charged capacitor and $C$ is the capacitance of the capacitor.
Hope this helps.
