Q:How capacitors store energy?
A:If you find capacitors mysterious and weird, and they don't really make sense to you, try thinking about gravity instead. Suppose you're standing at the bottom of some steps and you decide to start climbing. You have to heave your body up, against Earth's gravity, which is an attractive (pulling) force. As physicists say, you have to "do work" to climb a ladder (work against the force of gravity) and use energy. The energy you use isn't lost, but stored by your body as gravitational potential energy, which you could use to do other things (whizzing down a slide back to ground level, for example).
What you do when you climb steps, ladders, mountains, or anything else is work against Earth's gravitational field. A very similar thing is going on in a capacitor. If you have a positive electrical charge and a negative electrical charge, they attract one another like the opposite poles of two magnets—or like your body and Earth. If you pull them apart, you have to "do work" against this electrostatic force. Again, just like with climbing steps, the energy you use isn't lost, but stored by the charges as they separate. This time it's called electrical potential energy. And this, if you've not guessed by now, is the energy that a capacitor stores. Its two plates hold opposite charges and the separation between them creates an electric field. That's why a capacitor stores energy.