For big nuclei, the overall electromagnetic repulsion between the protons becomes sufficient (with a bit of help from quantum tunnelling/the uncertainty principle) to overcome the (strong) nuclear force that holds the nucleons together, and eject the alpha particle.
I haven't studied much of the details, but I think the reason alphas are emitted rather than individual protons is that alphas are quite a low-energy state because the two neutrons and two protons combine to fill the four lowest-energy nuclear states (one of each type of particle in each of the up and down spin states). Thus the mass of the alpha is less that the combined masses of its constituents, so less energy is needed to expel it.
The combined masses of the alpha and the nuclear decay product have to be less than the mass of the original nucleus. By expelling four nucleons, the magnitude of the (negative) nuclear force binding energy for the decay product reduces, but this is more than offset by the reduction in (positive) electromagnetic repulsion energy that results from it now having two less protons to hold together. In most cases, if it just spat out a single proton these sums would work out the other way round, so that doesn't happen.
The mass of the alpha is less than that of its constituents because in its case the negative binding energy of the nuclear force outweighs the electomagnetic repulsion energy of its two protons.
