In an exothermic reaction, the potential energy of the reactants is higher than the potential energy of the products. This difference in energy is released into the surroundings, usually in the form of heat or light.
The energy that is given off comes from the chemical bonds within the reactants. When these bonds break, they release energy. As new bonds form in the products, they require less energy than was originally stored in the reactants. The excess energy is then released, making the reaction exothermic.
For example, when you burn wood, the chemical bonds in the wood (reactants) break and release energy. The products (ash, smoke, and gases) have lower potential energy, and the difference is released as heat and light.