When Riding A Bicycle, Why Is It Easier To Balance While Moving Than At Rest?

It is a result of angular momentum.  A spinning object expends the least energy and experiences the least resistance when it's spinning around its center of gravity.  In the same way that the horizontal spinning of a gyroscope wheel keeps it from tipping, the spinning of the bicycle wheels creates a tendency for the bicycle to stay upright.  

Here is an excellent demonstration of how strong the force can be by the folks at MIT:
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A simple experiment you can do at home with very little prep is to balance a penny on its side on a table and bang on the table.  The penny will fall over.  Now, roll the penny across the table and bang on the table while it's rolling, It will roll merrily on its way.  The angular momentum of the rolling penny allows it to easily overcome the vibrations in the table.

Basically, the faster the wheels are spinning, the greater the tendency of the bicycle to stay upright and the less you have to shift your weight and make corrections with the steering to maintain your balance.  As the wheels slow down, the force is reduced and your dependence on on true balance and steering increase.

The bicycle is designed to balance itself. If the bike should tip, the frame forces the center of mass to move laterally. Inertia resists and is reacted at the road by a lateral force pushing the bike into the fall. As wheels can roll forward easily but not sideways, this force cannot push the rear wheel significantly sideways. At the front, however, the wheel is forced into alignment with the direction of motion. As the bike is tipping, the direction of motion is modified from straight ahead to slightly toward the falling side. Since the wheel can't be pushed sideways, it must align with this direction. The lateral force generated by inertia is then greater at the front than the rear and the bicycle yaws to steer it into the fall and so ends up over its base. Notice that the bicycle follows the center of mass and that is mostly the rider. This is why so little effort is needed to ride a bike.

When Bicycle is not in motion or is at rest then there are many other factors which are involved in balancing the bicycle. Like there is no gravitational, inertial or centrifugal, air, mass distribution. And yes torque also plays a part.  Well, the torque is produced by a force, for example an external force, the friction, or your own force. Torque produces movement, friction and when the bicycle is in motion these factors helps you to balance it easily.

While riding, if you start to tip, without even knowing you are doing it, you turn into the direction of the tip. Turning into the tip lifts you back up. Watch someone ride a bike for the first time, and they are wildly turning Left, Right, Left, Right, Left, Right, Left, Right, Left, Right.