What trajectory will the yo-yo follow?

1. Why will the yo-yo do this?

While on the end of the string attached to my finger the yo-yo's motion was constrained and it was forced into a circular orbit.

Once freed from that constraint the yo-yo obeyed a 'law of nature' and flew in a straight line.

The 'Law' here is Newton's First Law of Motion which says that moving objects continue to move at uniform speed in a straight line unless acted on by an outside force or forces.

2. What forces other than the constraining string were acting upon my yo-yo before and after I let it go?

Although the energy I put into whirling the yo-yo around my head modified some of their effects, two forces acting on the flying yo-yo were:
gravity and friction.