You're forming new bonds with whatever the oxidizer is (could be oxygen, could be flourine, could be many elements...), and those energies of formation are indeed the sole source of the combustion energy.
James' answer is excellent. For a rigorous treatment do a little research into 'Enthalpy of combustion' and 'enthalpy of formation' - fascinating subjects
Absolutely, energy of combustion comes from broken bonds between atoms and structures. Moreover, firmness of bonds which produce after combustion are another factor.