How does the conservation of angular momentum effect the stability of a bike?

Conservation of angular momentum is fairly similar to conservation of linear momentum. In the latter case, we learn that the momentum of an isolated system will remain constant, and that if you wish to change the total momentum, you must apply an external force to the system.

With angular momentum, the total momentum of an isolated system will be constant unless an external torque acts on the system.

The bike has angular momentum due to the spinning motion of the wheels (and to a lesser extent, the pedals). when we say this momentum is constant, we mean both in size and direction. The direction of the angular momentum vector lies along the axis of rotation - the axle of the wheel. So, as long as no external torque is present, the axle of the wheels will remain horizontal, and the entire bike as a result will remain vertical.

The faster the wheels spin, the greater the angular momentum, and the more torque needed to produce a change. This is why little children always find that once they get a good start on the bike, they can ride it without fear!