This has to do with an effect called gyroscopic precession. A spinning gyroscope will act as if any pressure applied to it's rim occurred 90 degrees around it's rotation.
So, pushing on the right handlebar puts pressure to the right (from the left) at the rear of the wheel and to the right at the front. Moving 90 degrees around the rotation of the wheel, tips the top to the right and the bottom to the left, leaning the bike to the right. The weight shift caused by the bike leaning causes pressure on the right side of the bottom of the wheel. Moving around another 90 degrees, this pressure causes the wheel to turn into the lean, so the bike makes a balanced turn to the right.
