A catalyst works by lowering the activation energy for a reaction. This means there will be a greater number of molecules with enough kinetic energy to react, compared to the uncatalyzed reaction.
When a reaction is not yet at equilbrium, this will make the reaction go faster. Since both forward and reverse reactions will speed up, it will not take as long for the two rates to become equal. The reaction reaches equilibrium more quickly.
Compare the decreases in activation energy for the forward and reverse reactions. They are exactly the same. Once the reaction has reached equiibrium, the rate goes faster, by the same amount, in both the forward and reverse directions. The net result cancels out, so there is no change in the equilibrium that has been reached. A catalyst has no effect on a reaction that is at equilibrium.