My intro to K-L divergence was in the context of maximum likelihood estimation with a misspecified model. The asymptotic expected value of a parametric MLE is the parameter value that minimizes the K-L divergence from the true distribution to the model. That means that when the parametric model contains the true distribution, the MLE is a consistent estimator (as is well known). But it also gives you a way of analytically finding the (asymptotic) bias of the MLE when the model does not contain the true distribution.