Angular Momentum/Precession

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The other day my physics teacher pulled some sort of magic trick that I'm sure many of you are familiar with. By some sort of witchcraft it seems that a bicycle wheel suspended from a string on one end of the axis won't fall if its spinning fast enough. I am befuddled.

After reading up on a few threads on here about angular momentum it has become apparent that the direction of angular momentum that we use the right hand rule for is just a convention for using it in calculation. I can understand that if you look at the torque applied to the angular momentum their directions seem to cause the wheel to precess, but something about this doesn't seem right. If you look at all the things that are happening to an infinitesimal piece of the wheel (dm if we were integrating) nothing pushes it sideways... I think. There's its linear momentum and the force of gravity. What is going on?!!? Can anybody help me out with this one?
 

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