Two atoms absorb thermal energy when joining together to form a molecule. What happens to that thermal energy?

Energy is released when chemical bonds form and it's absorbed when bonds break. The amount of energy needed to break a chemical bond is called the bond energy. The net energy absorbed or released by a chemical reaction is equal to the sum of the bond energies of all of the bonds that were broken minus the sum of the bond energies of all the bonds that formed. If this value is negative then more energy was released than formed and the reaction is exothermic. Conversely, if this value is positive then the reaction is endothermic.

In the situation you're describing there's a net absorption of energy. This energy went into breaking the bonds in the reactant molecules, freeing up the atoms to form new bonds. The joining of the two atoms actually released energy, but since it's less than what was absorbed the reaction is endothermic and it cools its surroundings.