If you will ask "why?" enough times, you will always hit a "this is an axiom", i. e. "we don't know, but we have experimental evidence for it". So with this attitude, nobody can ever understand anything.
If you need to calculate something in mechanics, you can very well imagine what will happen even before calculating, because you're so used to it from the everyday life. Couple this intuition with a good grasp of the theory itself and you can not only calculate things, but you can also make rough estimates right off the bat, and check whether what you calculated was nonsense . All in all, if you achieve this, you understand mechanics. You don't need to know what makes Newton's laws tick.
The intuition for electromagnetism is harder to get, but it's still perfectly doable. And so you can understand EM too.
However, in QM, there is little intuition (except for the math intuition). If you want to solve a problem, then you need to just "shut up and calculate" (popular motto amongst physicists, also misattributed to Feynman ). So that's what Feynman meant when he said that nobody understands QM, I think.
(Feynman is awesome. I pick up the book "Surely You're Joking, Mr Feynman" at least once a month, and I always have a good laugh, and maybe it affects me a bit in my real life too )