In the notation of this page, the entropy H(P) is best thought of as: "The mean number of bits to encode a member of P, assuming an optimal code." And the KL

In the notation of this page, the entropy H(P) is best thought of as: "The mean number of bits to encode a member of P, assuming an optimal code." And the KL divergence KL(P,Q) is probably best thought of as: "The mean number of WASTED bits if you encode members of P assuming that they had come from Q."