Question: My physics teacher told me the statement "The energy of a capacitor is stored in its electric field".

Now this confuses me a bit. I understand the energy of a capacitor as a result of the work done in charging it, doing work against the fields created by the charges added, and that the energy density of a capacitor depends on the field inside it. But how exactly do I interpret energy being stored "within" a field?

I saw that there have been similar questions asked here, but I wasn't able to understand the answers clearly, or found them unsatisfactory.

Answer: 
**What "is" an electric field?**

One way to look at the world is to see it as a cellular automaton; something resembling [Conway's game of life](https://playgameoflife.com/). Of course it's not so simple — the playing field looks different from observer to observer, everything is fuzzy etc. — but as a mental *model* it's quite productive.
Each little volume has a set of properties which interact with other properties and propagate with each "tick" (only) to neighboring cells, thusly establishing the speed of light.

Such properties are essentially a set of "influences" on various kinds of "particles"1: Charges are attracted, repelled or deflected. Matter is attracted. Quarks are glued together. These "influences" are what we call "fields" in a macroscopic view. **Fields are properties of points in space time** that describe how this point in space time interacts with "particles".

**How can a field store energy?**

One of these is the electrostatic field, for example between capacitor plates. It can accelerate charges, which means that it gives them kinetic energy. ***The field can do work on matter*; that is why we say it "contains" that energy.** Like all energy it is conserved: Charges flowing along the field will weaken it until its energy is gone. "How" exactly it stores the energy is a somewhat meaningless question: "How" does a missile store its kinetic energy? Fields and their energy are properties of spacetime volumes; that is all we can say.

While "fields" (i.e., spacetime properties) are "concepts" they do have a certain reality (beyond the fact that we can measure "their" effect, which is somewhat circular): Their energy actually makes the space heavy, although that wording is a bit awkward: Energy *is* mass, so it doesn't *make* anything heavy, it *is* heavy. If you want a bit of pop science, a magnetic field strong enough could conceivably be "heavy" enough to [produce a black hole](https://www.wtamu.edu/%7Ecbaird/sq/2014/08/21/what-is-the-strongest-magnetic-field-possible-is-there-a-limit/), unless something else happens first, like spontaneous pair production etc. There is no principle reason why this should not be possible with other fields as well.

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1 I'm putting "particles" in quotes because what we call "particles" is, at a closer look, a volume with special set of properties that just happen to perpetuate themselves, sometimes infinitely, sometimes not.
It's a model, created to capture what is seen in experiments. If we attribute the energy of an electromagnetic interaction to the fields, we get the right answer. We don't have an alternative that works. There is no deeper explanation.