Okay, Richard Feynman was an amazing dude, but I think he may have unintentionally spurred some confusion to non-physicists about our understanding of quantum mechanics.
You have probably heard his famous quote "If you think you understand quantum mechanics, you don't understand quantum mechanics." Now, if anyone understood quantum mechanics, it was probably Richard Feynman. And when he said this, he did NOT mean that physicists just have no goddamn clue how to deal with quantum mechanics, or what types of things are going on. We do know how to calculate all sorts of fun stuff that is going on in the quantum world (thanks to the almighty Schrodinger equation).
The big anomaly is that the quantum world does not behave like the macroscopic world. It is completely non-intuitive. In classical mechanics, objects behave exactly like we would expect. Balls roll down hills, pendulums sway back and forth, cars that run off the road crash into things, etc.
But, in the quantum realm, the world is turned upside down! Particles behave as waves, you can't measure both the momentum and position at the same time, particles in boxes can theoretically tunnel through walls. It's lunacy! However, we can show these things mathematically. Hell, these are problems we encounter in undergraduate quantum courses. They are relatively simple to show.
So, I hope that clears things up. Quantum mechanics is understandable in a sense. It is just crazy and we don't understand why quantum sized systems don't behave like people sized systems. If someone tells you they understand that, well, they are lying.
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