Why Quantum Electrodynamics works:
If I could answer that question with any certainty, I would be a rich man. Basically, QED is accepted theory because its predictions have been met to great experimental accuracy. For instance, some important experiment measured the magnetic moment of the electron to be, in some units,
1.00115965221
but the theory guessed it would be
1.00115965246
This was good enough for just about everybody, because nothing else could get closer. So QED works because it agrees with experiment. Why Nature reduces everything to probabilities and acts in the framework of QED (rather than something else) is a question that has not yet been answered.