What are electromagnetic fields made of?

When thinking about fundamental entities, it's quite easy to ask a question that, upon reflection, is contradictory. The questions of this kind take the form: What is [some fundamental thing] made of?

The contradiction here is that there can only be an answer if the fundamental thing isn't fundamental!

The electromagnetic field is one such fundamental entity. It's not made of anything else, it just is what it is. In the context of QFT, photons (real and virtual) are, loosely speaking, "excitations" of this entity. Real photons are associated with the long range propagation of energy and momentum, i.e., electromagnetic waves. Virtual photons are associated with the electromagnetic force, i.e., the Lorentz force, as well as evanescent waves, and near field antenna radiation.

Fields are more fundamental compared with particles (as fundamental as string). Particles, such as electron, are the excitation of Dirac fields (sorry for that). Possible related discussion from M Strassler which may be helpful. http://profmattstrassler.com/articles-and-posts/the-higgs-particle/360-2/ Some points are the following :" A field is something that

1.is present everywhere in space and time,
2.can be, on average,  zero or not zero, and
3.can have waves in it.
4.And if it is a quantum field, its waves are made from particles.

So for example: the electric field is a part of nature that is found everywhere. At any given point in space, and at any particular time, you can measure it. If it’s non-zero on average in some region, it can have physical effects, such as making your hair stand on end or causing a spark. It can also have waves — visible light is such a wave, as are X-rays and radio waves.

Ok, so, what is a particle?

A quantum field’s waves cannot be of arbitrary intensity. The least-intense possible wave that a field can have is called a particle, and it often behave in rough accordance with your intuitive notion of “particle”, moving in a straight line and bouncing indivisibly off of things, etc., which is why we give it that name.

In the case of the electric field, its particles are called “photons”; they represent the dimmest possible flash. Your eye absorbs light one photon at a time (though it typically waits for several photons to arrive before sending a signal to your brain.) A laser produces very intense waves, but if you shield a laser with a screen so that only a tiny fraction of the light gets through, you will find, if you shield it enough, that the light passes through the screen in little blips — single photons — all of them equally dim."