Why is it easy to balance a bicycle when it is in motion compared to when it is at rest?

Actually it is easier to balance a bicycle at rest then
when it is in motion. The  reason for this is the conservation of the angular momentum.
A bicycle has wheels that once in motion, they rotate. Each rotating wheel is generating
a non zero angular momentum.

If a person has done the
experiment of holding a wheel in hands, one will remember how easy it is to change the
direction of the wheel if it does not spin, but how difficult it is to move it once it
has start rotating. To change the angular momentum (in absolute value or just in
direction) one needs to apply a proportional torque (thus
force).

The same phenomenon applies to bicycle. The angular
momentum of the wheel is perpendicular to its plane, thus parallel to the ground. To
change its direction one needs to apply a very big torque to it. Hence even if you tilt
your position a bit to the left or to the right, you will not be able to change the
plane of rotation of the wheel once it has started to spin fast enough. This is the
explanation of why once you started to paddle and have a certain speed you will not fall
off the bicycle, and why it is more difficult to maintain the equilibrium on a static
bicycle.