Reason Behind Decrease In Potential Energy
The potential energy that changes during reactions is electrostatic energy (+/- attraction, +/+ or -/- repulsion) from the charged particles (electrons and protons) within the atoms. The theory is described by Coulomb's Law (involving the potential energy of charged particles):

http://guweb2.gonzaga.edu/faculty/cronk/CHEM101pub/energy.html

Generally, this comes from the attraction of electrons to nuclei and the repulsion between electrons and between nuclei. When atoms/molecules/the "system" becomes more stable, we will see either an increase in the attraction present (electrons are closer or less shielded from nuclei) or a decrease in repulsion (electrons or nuclei are further apart).

http://ch301.cm.utexas.edu/atomic/#bonding/bonding-all.php

More advanced theories will involve the changing of the allowed potential energy levels of the electrons either within atoms (atomic orbitals) or within molecules (molecular orbitals). Whichever theories are used, the net result is that there is more attraction and/or less repulsion in a more stable/lower potential energy state.

Interestingly, by the first law of thermodynamics, if the potential energy decreases, another type of energy must increase (so you don't end up with less energy than you started). It is normally thermal energy that increases and then is transferred as heat. This process ends up being one the most important ones in understanding why some processes happen and some do not.