No. It's commonly attributed to Einstein, but of course he didn't say that, because he knew better. There are things you just can't explain at a certain level. For example Feynman couldn't explain to a layman how magnets work, but explained why not: https://www.youtube.com/watch?v=wMFPe-DwULM.
But despite the claims to the contrary, he did explain magnetism to a layman, at a high level. Feynman's issue, seemingly a pet peeve of his by his tone, was the answering of a physics question asked by a layman that starts with 'Why'.
His long response actually covered just fine all the major points you'd expect in the answer, namely electromagnetic fields, alignments of the electron spins in the materials, etc.
His problem is that 'why' questions by non-experts are hard to answer because he doesn't know how deep to go before knowing he's satisfactorily answered with sufficient clarity and minimal hand-waving. And he did well in starting high and going a few levels deeper. Clearly he stopped before getting to a level requiring deep knowledge of physics.
And it's possibly his failure, not some intrinsic property of MAGNETS WORKING that's impossible to explain to laymen. In his own words,
"I can't explain that attraction in terms of anything else that's familiar to you."
"... because I don't understand it in terms of anything else that you're familiar with."
That can get you a fair distance in describing gravity (it's like the attract example, but much weaker) and the strong nuclear force (very strong attraction, but very very short range). That can then allow you to describe atoms, electrons and the protons in the nucleus attract each other, but [waves hands] not to the point they collapse into each other. And the nucleus, strong force generally overpowers the repulsion of protons.
And at this point a quantum mechanics for poets for chemistry works fairly well, it's not like any of that stuff makes intuitive sense anyway.
This doesn't contradict Feynman, of course, I can't say that I ever really understood magnetism, and I've also left the weak force out, but as I understand it you can wave your hands there and say it mediates beta decay. But the above does help you describe and understand the fundamentals of a lot of the universe, and much of what's important to you. Add Newtonian mechanics, which is pretty intuitive since we live in that world, and you're in fairly good shape. For that matter, a fair amount of special relativity can be easily added, the very basics of motion and the new definitions needed.
That really only answers the "what", though; it doesn't touch on the "why".
We can describe the attraction and repulsion, throw a precise mathematical framework around it, describe how two or more things might just be different aspects of (or viewpoints on) the same underlying phenomenon, even put names and behaviours to the particle-like things involved, but when it comes right down to it, nobody knows why any of it should result in forces that tend to move physical things closer together or further apart.
And that's okay, but it does mean drawing a line somewhere and saying "this is fundamental (at least for now)".
And it's safe to say no one really understood magnetism properly before quantum mechanics, qua Feynman.