I just watched the same video and it was one of the most insightful things that I have seen recently. Feynman is simply acknowledging that there are a few things in physics which are very difficult reduce to layman concepts. The other bit, about "why questions", is also quite interesting. I have come to believe that a lot of questions which sound very interesting, like "why do I exist?", don't really have good answers, and probably will never have. e.g. if it did turn out that we are living in a matrix, the question will simply reduce to "why does the matrix exist?" and soon. Thats what Feynman is trying to point out that "Why something happens?" is not a complete question unless we agree to some understanding beforehand. Otherwise you just go deep into the rabbit hole.
Listen carefully for the questions of the interviewer. He first asked "what is it, the feeling, between the two magnets," Feynman asks "what do you mean the feeling, what do you want to know," "I want to know what's going on," Feynman answered "well they repel each other," then the interviewer really asked "what does that mean, or why are doing that, or how are they doing that."
So Feynman tried to answer the "why" question. The shortest answer would really be "they just do it." There's no "purpose" it's just one manifestation of the electromagnetic force which is behind almost everything we see or feel. It's true, he didn't answer the "how" and that's why some viewers feel "injustice."
Fascinatingly, it's hard to answer "how" without knowing what he who questions already knows. Because the same electromagnetic force is behind almost everything we "see or feel," not just magnets.
If I'd guess the background of average person asking that question, the person would know that the electromagnetic force exists, as "I've heard the name," but not more than that. And he would be intrigued that he "feels" the repulsion between the two objects even if there are "no batteries," he'd actually want to know how it can be that the magnets "work" all the time and "at the distance." But even that isn't true, they don't "work," looking as a physicist, we do the work while pushing the magnets against each other to feel their repulsion, the force exists all the time. So what would be the satisfying answer?
I guess something like "the force you feel is everywhere, but you don't see that particular behavior. In these static magnets the tiny particles in the material can be easily nicely lined up that when you press them to each other you can feel the force differently than with the other materials, where the lining up normally doesn't happen or can't happen."
The interviewer also asked "how when we turn them then they stick to each other." That needs much more talk I'm afraid, unless you just say "they have poles" where you don't really explain the mechanism behind.
I've watched Feynman's videos where he explained some aspects of electromagnetism in a popular way ("that's how we don't fall through the floor" is the catchiest sentence in the talk), he was surely able to "explain the magnets" similarly (although not all the details in the short answer, he actually did a year long lectures for non-physicist students), but here he elaborated the problems in "why" questions.
Once a person already has decent knowledge, he can really enjoy the explanations like this: