The deeper reason why they do this, if you're interested, is due to their quantum spin value and the way that their wavefunctions are formed and added together. Bosons have integer spin, which gives them a symmetric wavefunction. These can be added together to get another symmetric wavefunction, which is a solution to the schroedinger equation of the system. Fermions, having antisymetric wave functions to start with, cannot have those wavefunctions added together in such a way that the resulting wavefunction is a solution to the schroedinger equation. I think.
Your question, by the way, is perfectly correct right up to the last sentence. Electrons only stick together in very special circumstances, usually at very cold temperatures, where they form cooper pairs. Otherwise they always repel.