What had Feynman meant when he told nobody understands Quantum mechanics? What do we mean by understanding Quantum mechanics?
I read the two existing answers (Anna; Valter Moretti) and I judge that whereas they are saying some correct things, they are not identifying correctly the issue which Feynman was pointing out.
The situation is that with quantum theory we have a very well-defined and detailed prescription for calculating, correctly and precisely, what a vast range of phenomena are like. In this sense we understand quantum mechanics very well. Feynman knew this as much as anyone. But he also knew that it is very hard, perhaps impossible, to present quantum theory in the style of `here is the state of the system at some initial time, here is the equation describing the evolution, and consequently here is the state of the system at some final time'. The formalism doesn't fall into that neat separation, and nor is there any other straightforward picture for the type of evolution it describes.
For example, in the calculation method famously associated with Feynman (the path integral approach), the integrals tell you the value of a quantum amplitude such as $$ \langle B | A \rangle $$ where $A$ could be some initial state of affairs (e.g. particles located at some given places, or having some given momenta and spin etc.), and $B$ some final state of affairs. The modulus-squared of this is (proportional to) the probability for that process. But the problem is that the further evolution of the system might involve a quantum superposition of state $B$ with some other possibility $C$. If this happens then one cannot say the process $\langle B | A \rangle$ happened on its own; one can only say that it is part of a larger process.
Thus one is led inexorably into the much-written-about puzzles of exactly what words like "measure" or "observe" mean. Also Bell's inequality indicates the impossibility of describing quantum systems as if each one can be characterized by a set of physical parameters local to itself. This hints at a rather subtle limit to the concept of reductionism, but one should be careful not to overstate that limit. In short, it was these sorts of questions that Feynman was alluding to in his comment.
What had Feynman meant when he told nobody understands Quantum mechanics ? What do we mean by understanding Quantum mechanics?
He probably meant that there is no inherent in our classical physics training, intuitive expectation of the behavior of matter in the quantum framework.
In the classical framework, we understand why the apple falls, once we know the force of gravity, and the behavior of all projectiles is easily understandable and intuitievely predictable. This is not true in the quantum mechanical framework.
He certainly meant understandable in terms of principles of classical mechanics (points of matter with masses, subjected to forces in a three-dimensional Euclidean space and satisfying deterministic evolution laws), since we understand very well quantum mechanics. That is evident from the fact that nowadays most technology is strongly based on QM so we can handle practically and, before, theoretically all notions and theoretical constructions of the quantum world. Exactly as we do with classical physics. In that sense we understand both.
The idea that classical physics is more understandable is an illusion in my opinion: we are only much more familiar with it and its theoretical description than quantum physics and relativistic physics.