Exactly, and that "why" is answered thoroughly, see the transcript.

The part of "how" is also there:

"If you're somebody who doesn't know anything at all about it, all I can say is the magnetic force makes them repel, and that you're feeling that force.

You say, "That's very strange, because I don't feel kind of force like that in other circumstances." When you turn them the other way, they attract. There's a very analogous force, electrical force, which is the same kind of a question, that's also very weird. But you're not at all disturbed by the fact that when you put your hand on a chair, it pushes you back. But we found out by looking at it that that's the same force, as a matter of fact (an electrical force, not magnetic exactly, in that case). But it's the same electric repulsions that are involved in keeping your finger away from the chair because it's electrical forces in minor and microscopic details. There's other forces involved, connected to electrical forces. It turns out that the magnetic and electrical force with which I wish to explain this repulsion in the first place is what ultimately is the deeper thing that we have to start with to explain many other things that everybody would just accept. You know you can't put your hand through the chair; that's taken for granted. But that you can't put your hand through the chair, when looked at more closely, why, involves the same repulsive forces that appear in magnets. The situation you then have to explain is why, in magnets, it goes over a bigger distance than ordinarily. There it has to do with the fact that in iron all the electrons are spinning in the same direction, they all get lined up, and they magnify the effect of the force 'til it's large enough, at a distance, that you can feel it. But it's a force which is present all the time and very common and is a basic force of almost - I mean, I could go a little further back if I went more technical - but on an early level I've just got to tell you that's going to be one of the things you'll just have to take as an element of the world: the existence of magnetic repulsion, or electrical attraction, magnetic attraction.

I can't explain that attraction in terms of anything else that's familiar to you. For example, if we said the magnets attract like if rubber bands, I would be cheating you. Because they're not connected by rubber bands. I'd soon be in trouble. And secondly, if you were curious enough, you'd ask me why rubber bands tend to pull back together again, and I would end up explaining that in terms of electrical forces, which are the very things that I'm trying to use the rubber bands to explain. So I have cheated very badly, you see. So I am not going to be able to give you an answer to why magnets attract each other except to tell you that they do. And to tell you that that's one of the elements in the world - there are electrical forces, magnetic forces, gravitational forces, and others, and those are some of the parts. If you were a student, I could go further. I could tell you that the magnetic forces are related to the electrical forces very intimately, that the relationship between the gravity forces and electrical forces remains unknown, and so on. But I really can't do a good job, any job, of explaining magnetic force in terms of something else you're more familiar with, because I don't understand it in terms of anything else that you're more familiar with."