A rotating bicycle wheel has angular momentum, which is a property involving the speed of rotation, the mass of the wheel, and how the mass is distributed. For example, most of a bicycle wheel’s mass is concentrated along the wheel’s rim, rather than at the center, and this causes a larger angular momentum at a given speed. Angular momentum is characterized by both size and direction.
The bicycle wheel, you, and the chair form a system that obeys the principle of conservation of angular momentum. This means that any change in angular momentum within the system must be accompanied by an equal and opposite change, so the net effect is zero.
Suppose you are now sitting on the stool with the bicycle wheel spinning. One way to change the angular momentum of the bicycle wheel is to change its direction. To do this, you must exert a twisting force, called a torque, on the wheel. The bicycle wheel will then exert an equal and opposite torque on you. (That’s because for every action there is an equal and opposite reaction.) Thus, when you twist the bicycle wheel in space, the bicycle wheel will twist you the opposite way. If you are sitting on a low-friction pivot, the twisting force of the bicycle wheel will cause you to turn. The change your angular momentum compensates for the change in angular momentum of the wheel. The system as a whole ends up obeying the principle of conservation of angular momentum.
Unfortunately, the gyroscopic precession of the wheel hanging from the rope is not explainable in as straightforward a manner as the rotating stool effect. However, the effect itself is well worth experiencing, even though its explanation is too difficult to undertake here. For more information, consult any college physics text on the subject of precession.