Why are marginal rates of substitution are identical at equilibrium consumption levels? According to wikipedia, "At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical". Why is it so?

What does "equilibrium consumption levels" here mean? I thought that equilibrium means that demand equal to supply. How is it applicable for marginal rates of substitution?
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.