Chemical bonds certainly "contain" potential energy, and the atoms want to move to a lower potential energy (become more stable).
When methane, #"CH"_4#, forms, the valence electrons end up in more stable (lower energy) C-H bonds.
These bonds are fairly strong, so methane is relatively inert.
However, if you add energy to the methane in the form of a flame or a spark in the presence of oxygen, some of the molecules will have enough energy to overcome an activation energy barrier.
Some of the #"C-H"# bonds will break.
The electrons can then enter an even lower energy state by forming #"C=O"# and #"O-H"# bonds rather than staying as #"C-H"# and #"O=O"# bonds.
So they “rearrange” themselves to form #"CO"_2# and #"H"_2"O"#.
The excess energy of 794 kJ/mol is released as heat, which we can then use to cook our food, among other things.
Thus, chemical bonds do not “store” energy. The energy for breaking bonds comes only when stronger bonds are formed instead.
This is the true driving energy for biochemistry, where cellular respiration provides energy by breaking the weaker bonds in carbohydrates and sugars and forming the strong oxygen bonds in carbon dioxide and water.
More energy is "available" because the weaker bonds are broken in favor of the stronger bonds being formed.
Many people say that ATP stores energy and releases it when the phosphoester linkage is broken and forms ADP. But it takes energy to break a phosphate group from ATP.
Rather, ATP provides energy when it breaks the weakly bonded phosphoester linkages and forms more strongly bonded glucose or fructose phosphate molecules.
Energy release comes from the net chemical reaction that produces new, more stable bonds to replace the less stable ones in the starting materials.