A journalist I am following on twitter just posted the question in the title of this post. I felt bound to try and give an answer with as simple concepts as I found meaningful. So, what makes a unstable particle unstable ?
One answer is this: a particle is unstable if there is a way, not forbidden by any physical law, to convert its rest-mass into other forms of energy. One may understand this by thinking of entropy: any system left free to evolve will do so in the direction of maximum entropy. So since a single particle state is a very low-entropy system, while the decay products of its disintegration have a multitude of possible configurations and a higher entropy, the system will naturally evolve in that direction.
Of course, the reaction must have a physically allowed way to occur: we cannot, for instance, make electric charge disappear, so an electron cannot turn into light quanta (it would also violate angular momentum conservation and a few other conservation laws, but that's beside the point).
Another possible answer, loosely connected to the above one, is that in physics anything that is not forbidden is compulsory. If the probability of occurrence of a reaction in a finite amount of time is not zero, that reaction will occur, if the system is given enough time. Similarly, if you play the lottery enough times, you are bound to win one day. But be prepared to live for a long time before that happens...
Maybe the most serious answer though is the following. Any quantum system is constantly subjected to quantum fluctuations, whereby virtual particles are emitted and absorbed. Take a muon: the muon is the heavy brother of the electron and, unlike the latter, it has a short lifetime (the electron is perfectly stable as far as we know). The muon constantly emits and reabsorbs photons, weak bosons, Higgs bosons: these are all the particles to which the muon "couples": the carrier of all the interactions to which the muon is subjected, by virtue of possessing some charges of the relative fields: electric charge and weak hypercharge.
Now these virtual emissions occur for very brief instants of time, such that the materialized virtual boson can exist without borrowing too much energy from the system: the rule is that the time of the virtual particle existence, multiplied by the energy needed to materialize it, needs to be of the order of the reduced Planck constant. This is Heisenberg's Uncertainty Principle
Note that if a muon emits a photon, even a virtual one, it remains a muon: only its energy may change in the process. However, if the muon emits a W boson, it turns into a muon neutrino (but we cannot yet speak of a real decay, until we clarify if the reaction undoes itself). Now take the virtual W boson emitted for a brief instant by the muon: its fate determines the fate of the emitting muon. If the W boson remains what it is before it gets reabsorbed by the muon neutrino, the muon will only feel a slight itch on the back and will not realize it has been turned into a muon neutrino for a brief instant of time.
But if the W boson also in turn "quantum-fluctuates" by emitting an electron -a virtual one also-, the W "becomes" an electron antineutrino (it is incorrect to say that a boson turns into a fermion: but here I am simplyfing things. You however well understand that what happens is that the W turns into the electron-electron antineutrino pair).
Now we're in trouble: the electron antineutrino cannot be reabsorbed by the muon neutrino, unless other quantum fluctuation reverse the whole chain in the exact inverse order (electron plus electron antineutrino merge into a W, and the W gets then reabsorbed by the muon neutrino, turning it into the original muon). So now the muon has become a muon neutrino, the W is no more, and the electron and electron neutrino are bound to live their different lives.
In a way, it is as if you used to lend your car to your friend Joe on Thursday evenings. Every Friday morning you get back your car intact. However, one night Joe crashes on a pole, destroying the car. The irreversible accident prevents you from getting back to your original state of every Friday morning: you are no longer a car owner.
It only remains to explain why the electron does not share the same fate of the muon in the reaction we have discussed above. Just as the muon did, the electron can also emit a virtual W boson, becoming an electron neutrino for a brief instant of time. And the W boson can also fluctuate into, say, a muon-muon antineutrino pair. However, no real accident can occur here, because at the end of these fluctuations the energy balance must return intact: and since the electron has a rest mass corresponding to a mere 0.5 MeV of energy, this is not sufficient to produce a real muon (which is 200 times heavier). The muon-muon antineutrino pair cannot, therefore, become real, and the reaction cannot develop: the virtual particle loop must close back into itself, and the electron returns to be itself.
Note also that the virtual W can indeed turn into an electron- electron antineutrino pair and make them real, without upsetting the energy balance. In that case, though, the antineutrino must annihilate (closing the loop) with the original electron neutrino (otherwise the two extra neutrinos would be carrying away energy, and energy balance would be violated). The result is just a quantum fluctuation which is absolutely indistinguishable from the one we discussed before.
Ian: I hope this helps!
What Makes Particles Unstable ?