If an electron is an excitation of the electron field, what causes the excitation to be stable? I won't pretend I understand even the basics of QFT, but from what I've heard about electrons, there are really two main ways of thinking about them. Quantum Mechanics describes an electron by a wave function who's squared magnitude gives the probability of finding the electron in a certain position or with a certain momentum. QFT, from what I understand, describes the electron as an excitation of the electron field. Both of these models describe the electron as some excitation of a mathematical field permeating space and time. My question is this:
If an electron really is described in this way (either using a wave function, or a field), what properties of the field, or of space-time itself, make it so the electron is stable (i.e. the excitation does not spontaneously decay, stop existing, or change) what causes the electron to continuously remain the same from one moment to the next)?
 A: 
If an electron is an excitation of the electron field, what causes the excitation to be stable?

I think the best way to say it is to take a tip from topological quantum field theory:
"Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory."
Note the reference to knot theory? Have a google on electron knot, and there's people out there saying the electron is stable because it is in essence a "knot" of field. This was discussed at ABB50/25 between Sir Michael Atiyah and other mathematicians and physicists. ABB50/25 was organised by Mark Dennis amongst others, who you can see mentioned here. 

If an electron really is described in this way (either using a wave function, or a field), what properties of the field, or of space-time itself, make it so the electron is stable (i.e. the excitation does not spontaneously decay, stop existing, or change) what causes the electron to continuously remain the same from one moment to the next)?

The "winding". Note that the positron has the opposite chirality to the electron. Chirality is something associated with knots. And when an electron and a positron meet, the result is typically two gamma photons. They aren't at all stable when their opposite chiralities cancel each other out. IMHO qftishard hinted at this in his comment, and you should bear it in mind if anybody tries to palm you off with "the electron can't decay because the theory says it can't". 
A: The electron is stable because there is no allowed process in the quantum field theory it can undergo that would lead to its decay. Its mass is the smallest among the electron/muon/tauon, so it doesn't have enough energy on its own to turn into one of those, and all other processes you could imagine are forbidden by conservation laws - either those of energy and momentum, or that of charges under the electromagnetic, strong or weak forces.
This has nothing to do with it being the "excitation of a field". For instance, the tauon is qualitatively the same, but its high mass allows a plethora of decay processes that the electron cannot undergo, and it is therefore highly unstable.
Note that I interpreted "stable" here to mean that the electron cannot decay on its own. If other particles are present, it might undergo electron-positron annihilation, or electron capture or something else.
A: 
There are really two main ways of thinking about [electrons]. Quantum
  Mechanics describes an electron by a wave function who's squared
  magnitude gives the probability of finding the electron in a certain
  position or with a certain momentum. QFT ... describes the electron as
  an excitation of the electron field. Both of these models describe the
  electron as some excitation of a mathematical field permeating space
  and time.

This understanding is generally correct, but the last line, “Both of these models ... permeating space and time”, is not a proper statement. In single-particle non-relativistic quantum mechanincs, or “first quantization”, electrons are eigen-states of a Hamiltonian, not a spatio-temporal field (in its proper sense). The quantum fields appear, in the precise sense, in many-body quantum mechanics or “second quantization”.

If an electron really is described in this way (either using a wave
  function, or a field), what properties of the field, or of space-time
  itself, make it so the electron is stable (i.e. the excitation does
  not spontaneously decay, stop existing, or change) what causes the
  electron to continuously remain the same from one moment to the next)?

The stability of the single-electron state is only due to the absence of interactions. In the non-interacting case, the electronic state (more precisely, the quantum state corresponding to a single electron) is an eigen-state of the Hamiltonian and therefore, remains stable. This is comparable to question on the stability of electronic orbitals in an atom < Why don't electrons crash into the nuclei they "orbit"? >; they do not loose energy or fall into the nucleus just because the corresponding states (or “orbitals”) are eigen-states of the atomic Hamiltonian. Being an eigen-state implies that the state will not be changed in the course of the evolution of the system.
Introduction of an interaction between electrons (like the Coulomb interaction) or between the electrons and other particles (like photons or phonons) would lead to an instability of such an electronic state and therefore, decay processes.
