I know electrons have a negative charge and that they are subatomic
particles made up of even smaller particles,
This is incorrect. Electrons are, so far as we know, fundamental particles which just happen to have a negative charge of -1 in elementary charge units as one of their properties.
They are not, so far as we know, made up of even smaller particles. It behaves like a particle that is not composite and is basically a zero radius point in space called a point particle, to the fullest extent that it is possible to test this experimentally. As explained in the point particle link:
[T]here is good reason that an elementary particle is often called a
point particle. Even if an elementary particle has a delocalized
wavepacket, the wavepacket can be represented as a quantum
superposition of quantum states wherein the particle is exactly
localized. Moreover, the interactions of the particle can be
represented as a superposition of interactions of individual states
which are localized. This is not true for a composite particle, which
can never be represented as a superposition of exactly-localized
quantum states. It is in this sense that physicists can discuss the
intrinsic "size" of a particle: The size of its internal structure,
not the size of its wavepacket. The "size" of an elementary particle,
in this sense, is exactly zero.
For example, for the electron, experimental evidence shows that the
size of an electron is less than 10^−18 m. This is consistent with the
expected value of exactly zero.
Fundamental particles (a.k.a. elementary particles), in general, are each one of a finite number of ways that quantum fields can have a local excited state that each behaves in a well defined way.
So far, the fundamental particles we know about are six kinds of quarks, three kinds of charged leptons (including the electron), three kinds of neutrinos, the W+ boson, the antiparticles of all of these particles, the Z boson, the photon, eight kinds of gluons, and the Higgs boson (each kind of quark comes in three colors and each of those can have left or right parity, each kind of charged lepton can have left or right parity, all neutrinos in the Standard Model are left parity and all anti-neutrinos in the Standard Model are right parity). There is also one hypothetical particle, the graviton, which a great many scientists (but not all) believe is an additional fundamental particle.
This is reality as we observe it, and the Standard Model does not provide any deeper explanation for it. Many extensions of the Standard Model, such as supersymmetry, propose that even more fundamental particles exist. But, science has not pierced successfully yet to a layer more fundamental than the Standard Model.
I am just very unsatisfied with not actually knowing what an electron
is and why it has a charge.
So are lots of scientists. But, they haven't come up with any better explanations. At best, many theoretical physicists would suggest that it might be related to M-theory (i.e. string theory) somehow or other. But, there is no realized, specific model implementing string theory that answers these questions in any meaningful way.