Note: The following tables describes what happens at various times in the movie. To see the times, right-click on this link and watch it on your own player. You may then easily reference the times.
Time (s) Description
0:00 - 0:04 Shows the two forces acting on the bike wheel: Fstring on wheel and FEarth on wheel
0:04 - 0:07 This animation treats the pivot point as the place where the string and axle join. The string is attached at the lever arm, so exerts no torque on the wheel. FEarth on wheel has a non-zero lever arm (shown in red as the vector r); the torque is shown as the unlabelled green arrow.
0:07 - 0:19 Shows the different perspectives of the wheel, so that you know which way the vectors are pointing.
0:19 - 0:25 The wheel as it stood either had to have an (unshown) force holding it up, or just be a snapshot (I take the former approach). How would the wheel move? The change in angular momentum &Delta L would have to point into the screen (the same direction as the torque). Using our righthand rule (RHR), we see the wheel falls down as we expect.
0:25 - 0:27 Put the wheel back in its original position.
0:27 - 0:37 Spinning the wheel does not change the torque as the level arm r and the force FEarth on wheel do not change. Once the wheel is spinning, what does change is that there is now a non-zero angular momentum L. We should think of this as the initial angular momentum Li.
0:37 - 0:45 We know that the change in angular momentum &Delta L is into the screen and Li is to the right. To find what will happen to the wheel (i.e. the direction of Lf) we change our perspective to a top-down view.
0:45 - 0:49 Graphical vector addition is performed to find Lf.
0:49 - 0:51 We know most of the angular momentum is assocated with the spinning wheel. We move the final angular momentum arrow back to the origin, and show the current angular momentum in green.
0:51-0:57 We see that for the angular momentum to end up pointing the right way, the wheel has to precess around to the vector Lf. We have shown how to use torque to explain which way the wheel will precess!

Questions:

Note that these questions are about somewhat more subtle issues. Read on if you are still confused or interested after following the explanation above.

If the wheel went the other way, wouldn't it eventually end up pointing the same way?

The wheel would make it around to the same point eventually. The short answer is that we are dealing with a short time interval, and the shortest time is obtained by going counter-clockwise as shown in the movie.

The longer answer: The point here is that we can only say "Δ L and &taunet are parallel" for small time intervals. Strictly speaking, that relation is only true when &taunetis constant, but for small times &tau is approximatly constant. We can imagine taking a very short time interval so &Delta L is small (recall &Delta L = &taunet &Delta t). Going clockwise would take a long time, whereas going counter-clockwise would not.

Therefore our explanation really does explain why the precession is counter-clockwise when viewed from above.

What about the fact that it is rotating counter-clockwise (as seen from above)?
Doesn't that mean that there are vertical components of L to worry about?

There are vertical components of L due to the fact that the wheel is precessing conuterclockwise. Using your RHR you should be able to tell that the vertical components of angular momentum points upward (the floor being the obvious direction for "down").

However, we saw quite clearly that the torque (as given by the RHR) was directly into the page. Therefore the change in angular momentum must also point into the page. But as soon as rotation starts there is a vertical component! How does this work?

The trick is that the wheel dips down, so the vertical component of the wheel's angular momentum is pointed down. The component gained from precession that points up exactly cancels this downward part. Thus the change in total angular momentum is still into the page, it is just that the wheel is tilted slightly. The diagram below should help explain how this works:

Don't worry too much about this: it is a fairly subtle explanation; typically the dominatant effects are the precession and the fact the wheel does not fall completely.