A person sitting securely on a rotating stool has his arms...

Since no external torque acts on the system about the axis of rotation, the angular momentum of the system must be conserved.

According to the law of conservation of angular momentum, $L = I\omega = \text{constant}$.

When the person folds his arms, his moment of inertia $I$ decreases. To conserve angular momentum, his angular velocity $\omega$ increases, but the total angular momentum $L$ remains unchanged.