According to Le Chatelier's principle, when we increase the concentration of a compound that participates in the reaction, the equilibrium will move. Therefore, if we add the conjugate base in an acid dissociation, the $[H_3O^+]$ will change, affecting the pH.
Work Step by Step
Considering the generic acid dissociation: $HA(aq) + H_2O(l) \lt -- \gt A^-(aq) + H_3O^+(aq)$ If we add $A^-$ to the solution, the equilibrium will try to favor the reactants; therefore, the reactants size will be favored. Since $H_3O^+$ is on the products side, it will be more consumed, changing the pH.