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Got two words for ya: gyroscopic action. Spinning gyroscopes (the wheels) spin on their axes (the axles). The spinning gyroscope can be moved without a lot of difficulty in any direction - except one that will translate its axis of motion. What that means is that a gyroscope resists any effort to change the way the axis of rotation points. This has a direct bearing on the stability of the bike. The spinning wheels are spinning gyroscopes, and they do not want to have their axes of rotation pointed in another direction other than the one they are in. In other words, the wheels don't want to "tip over" when they're spinning. And the faster they spin, the more they resist tipping over. The net result is that the bike will be a lot more stable at high speed than at low speed. When the bike is not moving, the rider will be able to stay up only by virtue of balance; he will get no help from gyroscopic action. There is quite a bit more to the issue of gyros and bikes. Steering a bike at speed is a challenge that requires the rider to overcome resistance offered by gyroscopic action. Additionally, in anything but straight line riding (on a flat road, and with no wind, etc., etc.), a bit of digging will have to be done to realize all the forces acting on a bike and its rider. A couple of links are provided to get you started.

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Q: Why is it easier to stay on a bike while moving but extremely difficult once it stops?
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