The situation is the following: in "ordinary" QM, the formalism
is entirely understood ; in quantum field theories (such as QED), even the formalism is not entirely understood, but well enough to do some calculations. In other words, we know that the calculations that are done, are mathematically somehow unsound, but this can be explained away by saying that it must be an approximation to something else (effective field theories).
But Feynman wasn't talking about the formalism, he was talking about the "physical meaning". This
is what is not understood, although several attempts with varying degrees of success have been invented. Nobody really knows what the mathematical objects in a quantum theory actually represent.
Some claim that it is just a mathematical tool which gives you statistical outcomes of experiments (in other words, that one shouldn't look for any physical meaning) - fine, but they can't come up with an explicit underlying physical mechanism ! Some (Bohr, with Copenhagen) claim that there IS no explicit physical mechanism, that all there is, is "statistics". This is essentially the "standard" Copenhagen interpretation: the quantum-mechanical formalism links statistically setups and outcomes of a "classical" macroscopic world, and there is no underlying explanation for this link. The formalism of quantum mechanics simply allows you to calculate the probabilities, but doesn't represent anything physical, because there IS nothing physical at that scale.
Others (such as me) claim that the formalism of quantum theory is to be taken seriously, and that it represents genuine physical quantities. These views are "many worlds" views, because you cannot avoid that way, to make a distinction between "the physical state" and "observed reality by an observer", which is so terribly weird.
Others think that the quantum formalism has something real to it, but that there is also an explicit "projection" mechanism. However, this usually introduces some clashes with relativity.
Still others think that there's something fundamentally wrong with the quantum formalism, although it makes correct predictions in many cases, for an ununderstood reason.
This is a discussion that goes on now for almost 80 years, and is usually referred to as the "measurement problem" of quantum theory. However, and that is the nice part: you don't need to think about all this to get the formalism working in practical cases, and in any case, it is a good idea to learn very well the formalism before delving into these issues.
So, the practical attitude to adopt when learning the formalism of quantum theory, is simply this: "quantum theory is a mathematical model which allows you to generate statistical predictions for outcomes of experiment, but for which no evident physical interpretation is known." From the moment that you try to do so, you delve into the problems of the measurement problem, which haven't really been resolved since about 80 years. This is what Feynman meant.