Formation of bonds and heat release

At the atomic/molecular level all chemical reactions are Quantum Mechanical phenomena where atomic and/or molecular electron orbitals of the reactants are being destroyed and new ones created in the reaction products. This is what you are correctly referring to as formation of bonds (although it has to noted that some bonds are also broken during chemical reactions).

These electronic transitions from Quantum Mechanically allowed energy levels to other, generally lower energy levels, are accompanied by the emission of electromagnetic radiation, usually referred to as photons.

What happens next depends a little on the actual reaction conditions.

Highly exothermic (i.e. releasing much energy) reactions will often emit much of the radiation directly, think for instance of the intense light emitted by burning magnesium.

But less exothermic reactions are often carried out in inert solvents (like water, ether, hydrocarbon solvents and many others) in which case the radiation is mostly absorbed by the solvent and the reaction products, leading to a temperature increase during the reaction. The energy of reaction has been effectively converted to Enthalpy (heat energy) of the solvent and the reaction products.

Contrary to @Gert 's answer, creation of photons from reactions forming and breaking chemical bonds (typically reactions between molecules) is quite rare.

When photons do occur, it can be (1) the result of a molecule, atom, or structure created or altered by the reaction being in an electronically excited state, which then decays, emitting a photon. There can be a delay ranging from microseconds to minutes (depending on a lot of factors) before the photon is emitted. Regardless, in fluids, the emission usually occurs away from the immediate microscopic neighborhood of the reaction.

A photon can also be (2) the result of incandescence, the radiation of heat. In the case of Magnesium burning, the reaction is so very exothermic that some of the remaining Mg is hot enough to radiate heat in the visible and UV spectrum. This is heat creating light, not light creating heat.

What typically happens in exothermic reactions is that the products, and other bodies in contact with the reacting bodies, receive this energy distributed as kinetic energy or excited rotational or vibrational quantum states. As such, this is the microscopic definition of heat. The heat is quickly shared with it's environment unless there is very low density of matter, such as in interstellar space.

A little more counterintuitive is the case of an endothermic reaction, which actually removes energy from the environment as kinetic energy or excited rotational or vibrational quantum states. In other word, the immediate environment gets colder.

Heat of formation (enthalpy of formation for an open system with mass transfer) is used to evaluate the thermodynamics of chemical reactions; e.g. the energy released when hydrogen combines with oxygen to form water vapor. This approach was developed long before the special theory of relativity was developed. We now know that the heat of formation is due to changes in the rest mass energies of products and reactants; if the heat of formation is negative, this is an exothermic reaction, the products have less rest mass energy than the reactants and the difference appears as increased kinetic energy of the reactants. A positive heat of formation is an endothermic reaction; the products have more rest mass than the reactants. This is the same concept as for a nuclear reaction but the changes in rest mass are much greater for the nuclear reaction; see discussions on the curve of binding energy in any basic nuclear physics testbook.