Why are marginal rates of substitution are identical at equilibrium consumption levels?
One explanation I could think of: MRS (which is equal to $MU_x/MU_y$) is equal to the price ratio $P_x/P_y$. In equilibrium, all consumers will consume where MRS = price ratio, and hence MRS will be "identical" across consumers.

I agree that this might not be a terribly good way of putting it.