How is energy released when a chemical bond is formed?
It is easier first to think of what happens when a bond is broken in an exothermic reaction. The energy is initially released into vibrations,  rotations and translational motion of the two fragments. This energy is then lost as these fragments collide with other gas molecules or with solvent molecules and so eventually ends up as heat. 

In exothermic bond formation the extra energy vibrationally and rotationally excites the new molecule and this is then lost via collisions with surrounding gas or solvent molecules and again ends up as heat. 

(If the new molecule were isolated in space (so no collisions) it would remain 'hot' until the energy is radiated away from vibrational levels and until it (radiatively) comes into equilibrium with the surroundings. This is a  slow process, by many orders of magnitude,  compared to collisions in solution or in a gas at normal atmospheric pressure )

In a very few types of reaction an electronically excited state is produced in the product and then photons radiate (as visible light) some of the energy away. This is called chemiluminescence.