Step-by-step explanation
In my honest opinion I believe that the yo yo toy uses the principle of simple harmonic motion coupled with the principle of conservation of energy
1. The principle of simple harmonic
Explains/describes it to and fro movement up and down and sides ways
2. The principle of conservation of energy explains how it stores energy in potential form(bounces to the hand ) and converts it to kinetic energy (bounces away from the hand).