How do breaking chemical bonds turn into kinetic energy
Breaking chemical bonds does not turn their potential energy into kinetic energy. All bond dissociation enthalpies are positive (for stable bonds that actually exist), which means that breaking them is an endothermic or heat absorbing process, and therefore they absorb kinetic energy (heat is a transfer of molecular kinetic energy).

What you might be thinking of is exothermic reactions, where the enthalpy of the reactants is lower than the products, and so the reaction releases heat, or increases the kinetic energy of the surroundings.

The short answer to your question is "because energy is conserved, it has to go somewhere, and heat and light are the only places it can go."

I'll try to think about a better way to explain it that isn't too far away from reality, but at the same time isn't too complicated. If I can, I'll edit this answer later to include it.