In addition to everything else that's been mentioned in this thread (or the combination of those) it was their broad appeal; they were viable to those who just wanted to rock and they were (after coming under Epstein's hand) acceptable enough to the masses. Once they were visible on a larger stage, it was their sheer charisma that attracted the masses. Their music justified their popularity (because it was exceptional) and from there, it all just exploded.
It's been said The Beatles were the salve the world needed to help heal from a generation of war and loss and a need ...