It's a little complicated, although not so complicated that it has anything to do with the spin of elementary particles.
The basic reason is due to perpendicular axis theorem. If you have a plate, there is some moment of inertia about the axis perpendicular to the plate, and a different moment of inertia about any axis in the plane of the plate. The perpendicular axis theorem says that the moment of inertia about the axis perpendicular to the plate is twice that about the axis in the plane of the plate.
Now, suppose you attach a rod through the plate, but at a bit of an angle, and then you spin the plate around this rod. As you can imagine, it'll take some effort to keep the plate spinning about the rod; it'll be rattling around trying to spin in a different way. Because you're not spinning it about any axis of symmetry, the angular velocity vector is not lined up with the angular momentum vector and so it requires some torque to keep the plate spinning around the axis you want. But if you're just throwing a plate up in the air, you can't exert any torque on it---it's just going on its own. So if the plate is spinning about some funny axis, it has to do so in a special way in order that the angular momentum vector lines up with the angular velocity vector.
So, draw a diagram of this lopsided plate turning around a rod. The angular velocity will have some component perpendicular to the plate and some component parallel to the plate so that the overall angular velocity is parallel to the rod. Let's suppose that the angular momentum vector's component perpendicular to the plate is exactly as long as the component of the angular velocity perpendicular to the plate. Because the moment of inertia in the plane of the plate is half that perpendicular to the plate, the length of the component of angular momentum in the plane of the plate is half the length of the component of angular velocity in the plane of the plate. As you can see, the angular velocity vector is not lined up with the angular momentum vector.
Now we decouple the plate from the rod so that it can also spin about an axis perpendicular to the plate. If we spin the plate about this axis backwards at half the speed it was originally spinning about this axis, the length of the component of angular momentum perpendicular to the plate is now half the length of the same component of the angular velocity. And so the angular momentum vector is now lined up with the rod that we're spinning the overall system around. But because the angular momentum vector and the angular velocity vector are parallel, there's no need to exert any torque on the system to keep it going. So we can remove the rod entirely and the plate will wobble in the air in a ratio of 2:1.
For a similar, but more complete, description (with figures), see "Feynman's Tips on Physics," Section 4-10, The Spinning Disk, and "The Feynman Lectures on Physics," Volume I, Section 20-4 Angular Momentum of a Spinning Body.
Mike Gottlieb
Editor, The Feynman Lectures on Physics
Coauthor, Feynman's Tips on Physics
The basic reason is due to perpendicular axis theorem. If you have a plate, there is some moment of inertia about the axis perpendicular to the plate, and a different moment of inertia about any axis in the plane of the plate. The perpendicular axis theorem says that the moment of inertia about the axis perpendicular to the plate is twice that about the axis in the plane of the plate.
Now, suppose you attach a rod through the plate, but at a bit of an angle, and then you spin the plate around this rod. As you can imagine, it'll take some effort to keep the plate spinning about the rod; it'll be rattling around trying to spin in a different way. Because you're not spinning it about any axis of symmetry, the angular velocity vector is not lined up with the angular momentum vector and so it requires some torque to keep the plate spinning around the axis you want. But if you're just throwing a plate up in the air, you can't exert any torque on it---it's just going on its own. So if the plate is spinning about some funny axis, it has to do so in a special way in order that the angular momentum vector lines up with the angular velocity vector.
So, draw a diagram of this lopsided plate turning around a rod. The angular velocity will have some component perpendicular to the plate and some component parallel to the plate so that the overall angular velocity is parallel to the rod. Let's suppose that the angular momentum vector's component perpendicular to the plate is exactly as long as the component of the angular velocity perpendicular to the plate. Because the moment of inertia in the plane of the plate is half that perpendicular to the plate, the length of the component of angular momentum in the plane of the plate is half the length of the component of angular velocity in the plane of the plate. As you can see, the angular velocity vector is not lined up with the angular momentum vector.
Now we decouple the plate from the rod so that it can also spin about an axis perpendicular to the plate. If we spin the plate about this axis backwards at half the speed it was originally spinning about this axis, the length of the component of angular momentum perpendicular to the plate is now half the length of the same component of the angular velocity. And so the angular momentum vector is now lined up with the rod that we're spinning the overall system around. But because the angular momentum vector and the angular velocity vector are parallel, there's no need to exert any torque on the system to keep it going. So we can remove the rod entirely and the plate will wobble in the air in a ratio of 2:1.