What happens in an exothermic reaction at the atomic level?
In a chemical reaction, bonds are broken and bonds are formed. For example, when elemental oxygen and hydrogen form water, we are breaking bonds in the elements and making heteroatomic bonds connecting hydrogen and oxygen atoms:
$$\ce{O2 + 2H2-> 2H2O}$$
To emphasize the bonds, we could also write:
$$\ce{O=O + H-H + H-H -> H-O-H + H-O-H}$$
In a thought experiment, we could first separate the elements into "isolated" atoms, and then recombine them to form water. Bond dissociation energies estimate how much energy it takes to separate bound atoms into "isolated" atoms. In the case of the example reaction, it takes less energy to break all the bonds of the reactants (one double bond connecting oxygen atoms and two single bonds connecting hydrogen atoms) than to break all the bonds of the products (four single bonds between oxygen and hydrogen atoms). As a result, there is an excess of energy that becomes available, resulting in an exothermic reaction.
What affects the amount of heat energy released?
The reaction follows a more subtle path than completely ripping molecules apart into atoms and then making new molecules. However, because enthalpy is a state function, the result in terms of reaction enthalpy is the same. So the only thing that affects the reaction enthalpy is the set of reactants and products.
The amount of heat depends on how you set up the reaction. If the temperature before and after the reaction is the same, if the pressure is constant and there is no non-PV work, it is equal to the reaction enthalpy. If the reaction is used to do work (electrochemical cell, steam engine etc), if it is a photochemical reaction, if it gives off light, or if any other of the conditions given in the previous sentence are not met, that will affect the amount of heat transferred.
Which form of energy is converted to heat energy?
The energy "comes from" the different energy of electronic states in the reactants and products. You can call this chemical energy, or a combination of kinetic and potential energy of the electrons. We see that electrons going from a higher energy state to a lower energy state give off energy when considering atomic spectra (sodium giving off a yellow light in the Bunsen burner, for example). In the case of reactions, electronic states are different between reactants and products not because they are in excited states, but because of the different configuration of nuclei giving rise to different electronic states.
How does this happen?
Unless the reaction gives off energy directly (by emitting electromagnetic waves - rather rare), the available energy gets transformed into kinetic energy of the nuclei (e.g. molecular vibration), and then is transferred to the surrounding by the common ways of heat transfer until the system is at thermal equilibrium again. If the reaction is endothermic, some of the activation energy (provided by molecular collisions) is not "returned to the pool" but instead used to get the electrons into their higher energy states in the products (compared to the reactants), leading to a net heat transfer from surroundings to the chemical reaction.