This is a qualitative answer:
The underlying level of nature is quantum mechanical. The classical concepts, like electric and magnetic fields, rise from this underlying level.
Take a charged capacitor. All the negative charges are on one face and all the positive on the other and there exists a classical electric field between the plates which has energy. This energy is the potential energy which separates the quantum mechanical charges, electrons have been moved from one side to the other and ions were created, and the energy stored comes from this operation.
If one takes an electron by itself, there is a charge, and an electric field associated mathematically with that charge, but there is no energy to be taken or given to the single electron because the charge is intrinsic. Thus the how of ensembles which give electric fields ,ends up to the basic question at the level of a single electron getting the answer "because that is what has been observed".
This always happens with physics because it is a science that uses mathematics to fit observations, and there are basic observations that are assumed. Particle physics has codified them in standard model particle table. So one can answer this question sequentially, peeling the onion of complexity, until one reaches the basic measured facts.
A more complicated argument can be made for the magnetic field, (one has to use the concept of moving charges, an observation) and in the end the answer to the last how will again end in the standard model table.
In order to move charges of equal sign that were separated by an infinite distance to some finite distance, you need to do work.
If you arrange an electric field $\vec{E}(x,y,z)$ in space by bringing in a large number of small electric charges (positive and negative) from infinitely far away to some arrangement, this takes work. For example, for a plate capacitor, you need to do positive work to collect positive charges close together on one plate, and positive work to collect negative charges on the other plate, and then negative work to bring the plates close together. The combined positive work from combining the charges on the plates will be larger what you get back from bringing the plates near each other, unless the plates touch and the charges cancel each other.
It turns out that the net work $W$ done to arrange the charges is equal to the integrated $dU/dV$ over all space:
$$ W = \frac12 \epsilon_0 \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} E^2(x, y, z)\, dx\,dy\,dz. $$
This equivalency of work (or energy) and the energy density of the electric field only exists in the above integral form. You cannot take an arbitrary cubic millimeter of space and somehow extract the field energy from it without affecting the field in the space around it as well.
As you have probably already done, you can verify that it works out for simple cases such as plate capacitors. Unfortunately, the general case requires mathematical knowledge (vector calculus) that you don't learn in high school. Once you have the vector-calculus toolbox and the description of electrostatics in vector calculus, the derivation is just a few lines. If you go and study physics in university, you'll learn this by the end of the first year.
I left out the effect of dielectrics ($\epsilon=\epsilon_0\epsilon_{\mathrm r}$). You can use the same reasoning if you start from an infinite dielectric medium, but it's a more tricky if $\epsilon_{\mathrm{r}}$ varies over space -- and unfortunately again too difficult for high-school level.
Best Answer
This question is deeper than you might expect. Neither energy nor an electric field is exactly what you might expect.
First, physics is a description of the behavior of the universe. It is not the universe itself. There are a number of different versions of physics - classical physics, quantum mechanics, relativity, etc - that describe different pieces of the behavior. None is perfect. We don't even know the total behavior. They are all pretty good.
It is common to think of energy as some sort of stuff that can live inside a moving object as kinetic energy. Or get stored in a stretched spring. It can be transformed but never created or destroyed. This point of view works in that it give the right answers. But energy isn't real. It isn't a thing in the universe. It is a tool to describe the universe.
We stand by the road and watch a car drive by. The car is moving and has a lot of velocity and kinetic energy. The drives sits in the car. He says the velocity is $0$. The seat is under him now. A while later, the seat is under him. It hasn't moved. If the velocity is $0$, so is the kinetic energy.
Both we and the driver are right. We are free to pick a frame of reference. As long as we stick to our frame of reference, we can use the tools of physics and get right answers. But you don't think of velocity as some sort of stuff that lives in a moving object and is gone when the car stops. If you are thinking about multiple frames of reference, thinking of energy this way doesn't work either.
On the other hand, if you do stick to one frame, it works just fine. So your teacher is right. It is a good, intuitive way to describe the universe. But it can get confusing if you ask too deeply just what it is.
Feynman compared energy to an accounting system. Here is a post that talks more about this point of view. Basic energy question
If we stick to one frame of reference, we always get the same answer for how much energy there is. We say that energy is conserved. We can say something similar about water. You can move it from place to place. You can evaporate it and move it into the air. You can condense it back to liquid. But you always have the same amount of water. Water is conserved. So thinking about energy in the same way you think about water makes it easier to understand energy.
The electric field is another tool of physics. This one helps sort out the forces between electric charges. Here is a post that describes it. In what medium are non-mechanical waves a disturbance? The aether?
Physicists get so used to thinking in terms of energy and electric fields that they forget they are tools and not the universe itself. They do talk about energy and electric fields as real things. They say the reason why that a moving object slows as it compresses a spring is that energy is conserved. They say the reason an electron accelerates is because an electric field exerts a force on it.
The purpose of these tools is to work so well, to match the behavior of the universe so well, that you can forget the difference. This helps make the universe understandable.
Responding to comments
People have brought up some valid points. This isn't something you need to pay attention to. You can do physics without it. So why did I bring it up?
How can a field be real, but not a photon? Or how can a charge be real, but not a field? Or field vs chair? How about the magnitude of the energy-momentum 4 vector? All good questions.
Let us agree that the universe is real and contains something. Physics is a model of the universe and contains mathematical descriptions of the universe that are different from the universe itself. If the descriptions are good enough, we need not care about the difference. It does matter if the descriptions don't match the universe in some important way.
There is a common view that energy is some sort of stuff that lives in objects or systems, and the total amount never changes. This view works from one frame of reference, but falls apart when you change frames. The OP was having problems understanding how this energy stuff could be stored in an electric field. It seemed that pointing out that energy isn't stuff would help with that.
But I have gotten a bit carried away in the comments. There is difference between what we think of as a chair and the thing that would exist even if we didn't think about it. But we don't need to worry about that difference when we sit down. Likewise we can use our models and do physics. Most of the time, getting nitpicky about what exists is getting away from physics and into philosophy. You might care about the difference if you are looking at the limits of where a model can be applied.
Let's just leave it at this: For every nonreal phsyics concept, there is a feature of the universe being modeled. Or at least approximated. If that feature is a repeatable pattern of behavior, or law, the question of whether is is "real", if it "exists", is more semantics than physics.