Thermodynamics: Catalysts make the rate of the forward reaction and that of the reverse reaction equally increase. Then why will the rate of the reaction increase?

A catalyst cannot change the Equilibrium constant, which is the ratio between rate constants. The scenario you described above cannot happen, since 20/10=2 and 30/20=1.5 20/10 should become 30/15 (or 40/20), causing a net rate of 15 (or 20) vs 10.

A catalyst changes the rate by lowering the activation energy. If you draw a reaction coordinate then there will be a hump between reactants and the lower energy products. The height of the "hump" is a barrier to rate. A catalyst works by lowering the height of this hump. It reduces the barrier of the change of reactants to products. Or in other words, it lowers the potential energy of the transition state between reactants and products. This allows the reaction to proceed faster. BUT the equilibrium concentrations are only dependent on the starting and ending energy. Since changing the height of the "hump" in the middle does nothing to alter the energy of the initial and final states there can be no change in equilibrium concentrations. A catalyst can help you reach equilibrium faster but cannot itself alter the equilibrium.