For example, an exothermic reaction will release energy to provide energy to overcome the activation energy. So, why the enthalpy change doesnt affect the rate of reaction?
The rate of a chemical reaction is directly dependant to the activation energy of that particular reaction. These two values are inversely proportional. The higher the activation energy, the lower the rate. As it has been mentioned though, the rate can be increased, in such cases, by increasing the temperature. All of the above is described as kinetics, which shows how fast the reaction will reach its end phase.
Enthalpy is linked with thermodynamics, which shows what kind of entities will be present at the end of the reaction. At what degree the reactants will turn to products.
I don't know your chemical background, so I tried to make this as simple as possible.
The enthalpy change of a reaction is just the difference in the bond ( potential energy) stored within the products compared with that within reactants. However, the activation energy is simply the amount of energy you need to break the bonds and start the reaction. To clarify the specific case you suggested: the only reason that an exothermic reaction would proceed faster than a hypothetical endothermic reaction of the same activation energy would be that the exothermic reaction is releasing energy that raises the temperature. However, if you could keep the temperature constant that would not happen and they would proceed at the same rate.