Ok, this is not a real direct answer to the question, but just a reaction to one of <PERSON>'s comments. My remark was too long to fit in a comment, so I put it in an answer, sorry for that. <PERSON> said he had a problem with one of the answers because : "yes, there are certain things that are axioms and not really derived so to speak, but there still must be some kind of intuition or justification. They don't just appear out of nowhere"
But I believe, they sometimes (often ?) do !
Actually, if we follow the historical train of thought on this matter, we should remember <PERSON> who was working to solve the UV catastrophe. He was desperatly trying to describe black-body radiation through statistical mechanics. Out of idea, he tried out the hypothesis that radiations were emitted in discrete bundles of energy E=hf. An idea that came (almost) straight up out of nowhere, as he admitted himself (at least, with absolutely no physical justification behind). He did not credit any physical meaning to it and considered it as mere mathematical trickery.
<PERSON>, acknowledging how well <PERSON>'s result was describing the experimental results, later declared that there was indeed a physical meaning in all this. He interpreted it as the statement that light could also behave as a discrete particle with a given energy. 682
Many consider this idea to be the beginning of quantum mechanics. <PERSON> later took back this idea and mirrored it : he said that if a "wave" such as light could be described as a "particle" then a "particle" like an electron could be described as a "wave". This equivalence is done through the famous relationship we are talking about, and that can be considered as a natural consequence of the E=hf relation, as it has been explained in another answer
So as you see, we can pretty well say that the hypothesis "E=hf" did come out of nowhere ! Definitely not an intuitive statement : it went against all intuitions of that time. Just an hypothesis that was working so damn well that we tried to put some sense into it... And came up with quantum mechanics. I believe it is something that you have to take first as a mathematical trick, later confirmed by experimental fact.
Trying to find an intuitive principle for something that, in its very core, is as counter-intuitive as QM is, imo, a desperate attempt. All those "paradoxes" and insane behaviours at the quantum scale have to come from something that is at least a little bit fucked up, wouldn't you agree ?
The idea that physics should be intuitive is something I generally disagree with. If it was, we would live on a flat earth, with a sun circling all around. That's intuition right there :D 682