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.