The energy release in chemical reaction comes from the difference in binding energies of the reactants and the reaction products. Thus, if we consider two atoms $A$ and $B$ that can join in a molecule, forming abound state $AB$ with energy $E_b$, then the energy conservation gives us:
$$
\frac{mv_A^2}{2} + \frac{mv_B^2}{2}=\frac{mv_{AB}^2}{2}-E_b
$$
or, if the energy is release as electromagnetic radiation (photon):
$$
\frac{mv_A^2}{2} + \frac{mv_B^2}{2}=\frac{mv_{AB}^2}{2}+h\nu-E_b
$$
That is the collision of the two atoms may result in them forming abound state, with the product having higher kinetic energy than the reagents:
$
A +B = AB + E_b.
$
(The energy of the bound state $AB$ is negative: $E_{AB}=-E_b$.)
The molecule may also dissociate into atoms (for stable molecules it rarely that it happens spontaneously, but it can be hit by another atom, providing the energy):
$
AB+E_b= A + B
$
As a reversible chemical reaction can go both ways, $A + B \leftrightarrow AB$, eventually a balance is established, where the number of new molecules becomes equal to the number of molecules that dissociate, and the reaction stops. Thus, in a closed system, although on the microscopic level the reaction would continue going infinitely in both directions, it stops on the microscopic level. If a reaction is maintained at constant temperature (isothermic), the exotermic reaction will result in excess energy that would be lost to the environment, and it will go till it cannot generate any excess heat anymore (in which case it reaches a kind of equilibrium described above).
Remarks:
- @JohnDarby correcly points out that the excess heat can be described as a mass defect. This description works well for nuclear reactions, but is less convenient for chemical ones, since the binding energy is much smaller than the rest energy of the reactants and products. Still, $E=mc^2$ applies here as well.
- As a crash course on chemical reactions for physicists I recommend Introduction to Physics and Chemistry of Combustion: Explosion, Flame, Detonation by Michael Liberman.