There is a lot more to this question than you might expect.
You can argue that an electric field is just mathematics. Is it even real? How is energy "stored in an electric field"? talks about that.
But there is something physical going on behind the math. The point of the math is to describe how charges interact with each other.
In classical physics, some particles are charged. That is, they exert forces on each other. If you place one near another, they will accelerate away from each other (two electrons) or toward each other (an electron and a proton). By contrast, an electron and neutron do not exert forces on each other.
The laws are relatively simple as long as the charges hold still. If one charge moves, there is a time delay before the other charge notices the change. The math is simplified if the problem is divided into two parts.
In part 1, a charge creates or has an electric field. The field is a mathematical description that fills all space. It describes how strong the force would be at a point if another charge happened to be there. If the charge moves, the change in the electric field spreads out at the speed of light. A moving charge also generates a magnetic field. These fields carry the "news" that the charge has moved.
In part 2, another charge feels the force of the electric and magnetic fields.
Maxwell's equations are the math that describe all this. For many years, these equations told us everything known about charges and electromagnetic fields.
If you set up some charges and solve Maxwell's equations for that charge distribution, you find the electric field. If you set up a vibrating charge, Maxwell's equations tell you oscillating electric and magnetic fields spread out like a wave. We call this light. If light strikes a charge, the charge feels an oscillating force. If this happens to be in a receptor in your eye, you see the light.
Maxwell's equations describe all the electromagnetism we see in classical physics. They describe a few things we don't see. You can set up empty space and solve for the electromagnetic fields. You can find light that is consistent with this. It propagates from the infinite past to the infinite future with nothing create it. This is a useful approximation for starlight that has left its star far behind. But all light (classically) is caused by the movement of charges.
Likewise, you can imagine magnetically charged particles and figure out the resulting fields from Maxwell's equations. We don't know of any reason why such magnetic monopole particles can't exist. But searches haven't turned up any.
So far, the fields could just be a mathematical trick. Charges and forces are real. But fields? See In what medium are non-mechanical waves a disturbance? The aether?
You can see why fields are a useful trick. We say that light in a distant star generates light. Years later, the light hits the lens of a telescope. We can understand how charged particles in the lens redirect the light. Given the shape of the lens, light can be made to converge toward a focal point. A detector at the focal point can respond to light in a way that generates a permanent record.
It is a lot harder to describe all this in terms of vibrating charges in a distant star setting up vibrations years later in the charged particles of a telescope lens that are just right to set up more vibrations in a detector.
But in quantum mechanics, point particles turn out to be too simple a description. Electrons are sort of like particles and sort of like waves.
It turns out that light is sort of like a particle too. This is a photon. How can a red light photon be different from a blue light photon? It is a little harder to dismiss a particle as a figment of math.
