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: 
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.
If you look at our equations for energy, you will see that they typically settle into two major categories:

* Energy associated with intrinsic properties of a "thing"
* Energy associated with the relative positioning of objects in the system.

A baseball flying through the air has some energy associated with its intrinsic nature. It has a kinetic energy. It has some chemical energy. But it also has some energy associated with its position - gravitational energy. Now at first glance it seems like this could be treated as a property of the ball, but let's do a thought experiment. Let's have Bizaro shove the Earth away. Maybe he sends it careening into the sun. That's a good supervillain ploy.
Did this change the gravitational potential of the baseball? It certainly did. But the baseball itself didn't change. So if the baseball didn't change, but its gravitational potential did, then it's pretty unreasonable to consider it to be a property of the baseball. It's a property of the Earth-baseball system.
The electric field is the same way. An electron is an electron, no matter where you put it. Its intrinsic properties do not change. But if you have two electrons, their *relative* relationship leads to a potential energy.

This can get confusing, because we regularly look at multiple levels of detail. At a high level, a capacitor is a "thing" that has an intrinsic property "potential energy due to charging." But if you look at it at a lower level, you see that that "potential energy due to charging" was actually a property of the relative positions of electrons in the capacitor. And this leads to an understanding of "parasitic capacitance," which is what happens when you accidentally bring two wires close together, and the relative positions of the electrons in both wires results in a capacitive effect you may not have planned on when building the circuit.

Realizing this is just a back and forth of modeling and approaches, rather than physics, is helpful. I know I was always very frustrated when I heard the term "phonon." A phonon is a particle of sound.... which is absurd because I know that sound is a wave mechanic, not a particle mechanic. But I also am wrong. Sound isn't a wave mechanic or a particle mechanic. Sound is sound. I associate a mechanic to it to aid in solving equations. And, as it turns out, there are some times where it is very effective to treat the energy of sound as an intrinsic energy of a "phonon" that propagates through space.