How does binding energy "manifest" itself?
If you want to break a bond you first need to excited the right degrees of freedom so the molecule can move along the reaction coordinate of interest. In other words, the correct vibrational modes need to be excited. At equilibrium distance this will be reflected in the kinetic energy of the atoms. As you move towards the dissociation limit this kinetic energy gets transformed to potential energy, much like you give a ball a push to move up a hill. Any excess energy (if you excite more than binding energy is required) will result in kinetic energy after the dissociation.

Forming bonds happens in reverse: Two fragment come closer and convert their potential energy into kinetic energy.

After such an reaction (bond forming or breaking) several options can happen:


the excited states go into lower states by emitting photons (e.g. fire, chemoluminescence)
the energy dissipates into other degrees of freedom (e.g. by momentum transfer in collisions with other molecules)
the reaction may happen backwards and you get your initial educts again
(there are probably further options ...)