As the other answers have mentioned, a simple A-B bond dissociation reaction is significantly endothermic and does not happen spontaneously. A single molecule may break apart and its pieces may be scattered if energy is introduced in the molecule (for example, a collision with a very energetic target or the absorption of a photon may trigger strong vibrational excitations which end up breaking the molecule apart). It is, however, also possible for an unstable molecule to spontaneously rearrange into a more stable isomer, and part of the released energy may cause the rearranged molecule to shake apart.
In any event it's a case of causing a vibration to stretch atoms strongly enough that the bond between them snaps and the pieces fling away, not unlike a rubber string being pulled until it tears. Perhaps this video may help clarify what's going on, though it doesn't specifically show reactions occurring.
Edit: I somehow had completely misread the question as $\ce{A +B -> AB}$ instead of $\ce{AB -> A +B}$, so I apologize for that. My previous answer isn't actually an answer, but I figure I might as well leave the information below in case anyone is interested:
As far as I understand, most often the chemical potential energy released when two molecules react exothermically turns into vibrational potential energy in the bonds of the product molecule/entity for a small amount of time, and this vibrational energy is quickly dispersed into other molecules by collisions (timescale of a few picoseconds), turning mostly into rotational and translational kinetic energy dispersed over several molecules, as vibrational excitations are "frozen out" at temperatures close to ambient.
If the two molecules meet in an absolute vacuum and can react exothermically but have nowhere to dump the energy, then the molecules actually have a large chance of not reacting at all, even if it's extremely spontaneous! The only way they can meet and stay bound is if, during the short time they are in proximity, a low-probability event may occur where the reaction transition state emits a photon carrying away most of the releasable chemical potential energy, otherwise the molecules will just fly apart again. This is a very important consideration in astrochemistry. You can read more about this specific topic in "The Physics of the Interstellar Medium", 2nd edition, 1997 J.E. Dyson and D.A. Williams (IoP), chapter 3, section 3.4.
Actually, I should clarify that the second paragraph is only really valid for reactions of the type $\ce{A +B->C}$. A reaction of the type $\ce{A +B->C +D}$ can happen more easily as one of the species being formed can also act as a dump for excess energy.