For more emphasis let me take another generic example: a person sitting on a swivel chair, arms outstretched, in each hand a weight. For simplification let me look only at the weights, disregarding all the other mass in the setup. When this setup has an angular velocity and the person contracts his arms the angular velocity increases.
As I understand it, your question is about the instantaneous linear velocity of the weights. That instantaneous linear velocity changes, and you wonder how that happens.
The image shows the trajectory of one of the weights during contraction. It's an inward spiral.
The fact that it is an inward spiral makes all the difference. For comparison, when the motion of the weights is along a circle then angular velocity will not change. The angular velocity doesn't change because at all times the exerted force is at right angles to the instantaneous velocity.
In the image the dark arrow represents the actual force. This force is at all times towards the center of rotation.
The two lighter gray arrows in the image show how the actual force can be decomposed into two perpendular components.
- a component at right angles to the instantaneous velocity
- a component parallel to the instantaneous velocity
This shows why the velocity of the weight increases: there is a force component parallel to the instantaneous velocity.
This is the key point:
When a rotating systen contracts the angular velocity increases because the centripetal force is doing work.
I'm aware of course that in many physics textbooks it is stated: "when a rotating system contracts the angular velocity increases because angular momentum must be conserved." However, that way of looking at it is problematic.
Compare the case of a cannon firing a projectile. When the cannon fires the projectile is accelerated because of the explosion of the propellant. It would be very weird to state: "The cannon has a recoil, and since linear momentum must be conserved the projectile must fly off." (Also, how would you then explain the recoil? You would have to turn around and say: "The projectile flies off, and conservation of momentum makes the barrel recoil.")
When something changes you identify the cause: when a rotating system contracts the angular velocity increases because the centripetal force is doing work, increasing the kinetic energy.