Newton found that an object in motion tends to remain in motion, in a straight line and at a constant speed, unless it is acted upon by a net force. Today, we call this observation the law of conservation of momentum. The momentum of an object is the product of its mass and its velocity.
There is an equivalent law for rotating objects. A rotating object tends to remain rotating with a constant angular momentum unless it is acted upon by an outside twisting force. The definition of angular momentum is more complex than that of linear momentum. Angular momentum is the product of two quantities known as angular velocity and moment of inertia. Angular velocity is merely velocity measured in degrees, or radians-per-second, rather than meters-per-second.
Moment of inertia depends on both the mass of an object and on how that mass is distributed. The farther from the axis of rotation the mass is located, the larger the moment of inertia. So your moment of inertia is smaller when your arms are held at your sides and larger when your arms are extended straight out.
If the motion of a rotating system is not affected by an outside twisting force, then angular momentum is conserved for this system, which means that the angular momentum stays the same.
A person sitting on a rotating chair or stool approximates a system in which angular momentum is conserved. The friction of the bearings on the chair stem serves as an outside twisting force, but this force is usually fairly low for such chairs. Because angular momentum is conserved, the product of angular velocity and moment of inertia must remain constant. This means that if one of these factors is increased, the other must decrease, and vice versa. If you’re initially rotating with your arms outstretched, then when you draw your arms inward, your moment of inertia decreases. This means that your angular velocity must increase, and you spin faster.
The conservation of angular momentum explains why ice skaters start to spin faster when they suddenly draw their arms inward, or why divers or gymnasts who decrease their moment of inertia by going into the tuck position start to flip or twist at a faster rate.