In the case of a capacitor, it's better to think of the energy stored in it as the work done charging it up.
You have to do work against the repulsion of the charge that's already on it. The more charge on there, the more work you have to do.
When charged, the energy is stored in the electric field, and when you connect the plates together in a circuit, the capacitor does work pushing the charges around the circuit as it discharges.
The energy is released.
Although a "direct" current doesn't flow through a capacitor, charge can flow onto and off the plates. This constitutes a current. If you have a circuit with alternating current, the capacitor conducts a current with no problem at all, first pushing charge onto one plate, and then onto the other.
The larger the value of the capacitance, the more charge can be stored on it (at the same V) so the bigger the alternating current can be for the same volrtage. This is why the "resistance" (correctly called reactance) of a capacitor to AC goes down as you increase its capacitance.