What are quantum fields made up of? If quantum fields are mathematical entities made up to explain nature, what they explain is definitely something physical and is made up of something. So why can’t there be an answer to what these mathematical quantum fields are made up of?
I mean, if physicists are making them up, then they definitely would have certain criteria as to what these mathematical fields are made up of, won't they?
 A: "The book of nature is written in mathematics", said Galileo. It is hard to sing a song about a book that cannot be read; here is a song.
Quantum fields are made up of quantum oscillators, an infinity-of-infinities of them. These oscillators are little gadgets, everywhere, that spew out and consume quanta, the building blocks of our world (the notes/tones of a song?).
Their understanding has been around for virtually a century (Jordan), but still hard to intuit: that's why it takes years of physics training to handle them blind.  These quanta are photons, matter particles, gravitons, and such.
They are more physical and may produce more accurate answers than engineering and any other physical theory. As a matter of fact, most of the engineer's/layman's intuition about the physical world is an elaborate empirical but flawed summary of quantum fields, and a real challenge is how that arises out of the more-real-than-real quantum fields. What you/we see in your day is but a warped image/bluff of collectivities of them, but you/we are spoiled since birth to assume it is more real.
As Steve Weinberg,  the late  (just) superhero of understanding such quantum fields said, "The universe is an enormous direct product of representations of symmetry groups”.  These representations are quantum fields.
In principle, they undergird everything, but working out how our cattywampus macroscopic world arises out of them, "emergence", is sometimes a challenge, and  rarely not. But there are hardly credible, complete, efficient, logical alternatives to them.
A better song is this one.
A: I'm going to take a very different position than Cosmas Zachos' amazing answer. That doesn't mean I think the answer is wrong, but rather that I believe there are multiple quite different answers to this question because it is not really a physics question, but rather a philosophy one. Hence, I'm bringing a complementary point of view.
My point of view: not only don't we know what quantum fields are made of, but we don't even know whether they exist (whatever "existing" means) and we don't really care. The point of physics is not to describe how the Universe actually is, but rather to say what it is like.
Pick Newtonian gravity, for example. It tells you that there is a classical field permeating spacetime, which we dub gravitational field, that produces a force between any two bodies which goes like the product of their masses divided by their distance squared. Is that how the Universe actually is like? Not necessarily, but for a wide range of phenomena it pretty much works as if that was the whole picture. You can get away with not knowing a thing about general relativity (GR) for quite a while. Alternatively, you can also formulate Newtonian gravity geometrically, which is how we recover it from GR in the weak field limit, and then conclude that there is no force, but instead spacetime is curved and gravity is just that.
Which one is the true nature of the Universe? One could argue curved spacetime because of GR, and then another one might start arguing about how is the nature of whichever theory lies below GR, or alternative gravity formulations. Which one of them is correct? There is no way of knowing, and albeit a fascinating problem, it is not a physics problem.
Quantum fields are pretty much the same thing. We don't observe the fields directly. What we can measure are quantities like cross sections or decay rates, which are numbers that tell us properties about how "particles" scatter off each other or how fast they decay (the quotation marks are because we interpret field excitations as particles, but the fundamental quantities are really the fields). Since we are not measuring the actual fields, we don't really have a way of knowing if they are there. The whole point is: the Universe works in a way which fits incredibly well with what one would have if the fields were actually there. We don't know whether they are, but it is really similar to if they actually were.
As I said, this is more of a philosophy question, which I'm not really an expert on. The position I'm taking is closer to what is known as anti-realism (where I'm skeptical about whether we can say if something we don't measure really exists), while Cosmas Zachos' answer is closer to a realist point of view (which is closer to the notion that if the theory is using such a concept, it must exist in some way).
There is much more to these ideas than I'm putting in this comment, but as I said I'm not an expert. What is worth pointing out is that we often change our mind when talking about different theories: the same physicist might say that GR implies that spacetime actually is curved (more realist-like), while denying the wave function is anything more than a mathematical convenience (anti-realist).
In summary, we don't know what they are made of, and attempting to answer the question involves making a philosophical choice. What we do know is that the Universe behaves just as if these things exist to a gargantuan precision. Maybe one day we'll find a deeper theory that replaces QFT and explains what the fields were made of, maybe we'll be stuck with it forever.
A: Quantum fields are mathematical constructs that represent quantum degrees of freedom. In relativistic quantum field theory, a quantum field can be used to represent an indefinite number of identical relativistic particles. Furthermore, quantum fields allow us to encode interactions that are local.
In the Standard Model, some particles (such as the electron) are "fundamental," and so the field associated with these particles are also "fundamental", meaning they aren't made of anything deeper that we know of.
This doesn't mean that there is not a deeper level of reality beneath the Standard Model, out of which our current set of fundamental fields emerge. This could be a different quantum field theory, or string theory, or something else. At the moment, we don't know whether this deeper level of reality is there (though most physicists think there is physics beyond the Standard Model; at least dark matter, perhaps dark energy, and also quantum gravity).
So the provisional answer is that some quantum fields are made of other quantum fields, and others are "fundamental", meaning (as far as we know now) there are no deeper explanations beneath them.
However, there are other mathematical constructs that can represent the same physics. More precisely: there are frameworks to compute scattering amplitudes - the results of relativistic quantum scattering experiments --- where the fields do not appear at all. Furthermore there is an enormous degeneracy in quantum field theory, where fields can be redefined but not change the physics. One way to say this, is that the fields are dummy variables in a path integral, and so the fields themselves can be transformed in an infinite number of different ways without changing the results of a physical observable. So I would avoid attaching too much "reality" to the notion of a quantum field, even though the formalism is incredibly useful and powerful.
In my opinion, a much more interesting question than whether fields are real, or what they are made of, is whether fields are useful. The answer is undoubtedly yes.
A: I'm going to give a bit more literal of an answer with a bit less story since that's my taste. What we call a quantum field is an operator as a function of position and/or time, i.e. at every $x, t$ there is an operator $\hat{\phi}$. Just like other operators, this operator acts on elements of the hilbert space. In mathematical physics, they are even more precise and formulate the fields as operator-valued distributions.
That is the simplest quantum field; in general it can have multiple operator components. For example, in QED the vector potential $A^\mu$ has components $A^0, A^1, A^2, A^3$ which are each operators. It is a quantum field.
We call them fields because in physics, the word "field" is used for (possibly time-dependent) objects which depend on position.
These mathematical objects are part of our theories which we use to make predictions for experiments like the ones at the Large Hadron Collider. However, we have in no way proven that they are fundamental to reality - they could just be part of our human formulation of (our best progress so far at) the fundamental theory of nature. So there is a sense in which we don't even know that they really exist, much less that they are made of one thing or another.
