While this is a short video the underlying physics of what is demonstrated is quite complex. Ask the students to work in small groups and draw the free body force diagrams of the three cases. From the diagrams they should then be able to give you an explanation of why it is possible to get 3 different types of motion by applying same magnitude of tension to the rope. As you have already guessed it is the angle of the rope to the hub of the “yoyo” that is the important factor.
So the pivot point for the “yoyo” is where it is in contact with the floor. Let us start with the simplest situation, no rolling motion. If you look closely you will notice that the angle of the rope is such that the force it provides is along a line from the pivot point to a point on the surface of the hub where the rope loses contact with the hub that it is wound around. So the force is in the direction of the lever arm and there is no rotation.
Then look at rolling away. The lever arm is again the line from pivot point to hub where the rope leaves the hub, but the rope is now more vertical. If the force supplied by the rope is broken into its component parts (you may have to draw a diagram) with one part in the direction of the lever arm and the other component at ninety degrees to the lever arm then you find you have a clockwise torque on the “yoyo” and it rolls away from you.
Using the same logic examine the situation where the “yoyo” rolls towards you. The angle of the rope is more horizontal. The point where it leaves the hub is lower so the lever arm is now almost vertical but the rope is almost horizontal and the force is almost perpendicular to the lever arm providing a torque that makes the “yoyo” roll in an anti-clockwise direction.