Lagrange's theorem intuition

Question

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation translates into: [G:H]/|G| = |H| (the number of equivalence classes, represented by y, divided by the cardinal of G, represented by x gives the cardinal of H). Where is the analogy flawed?