Another great example is Terry Tao's alternate proof of the Szemerédi graph regularity lemma. He uses an information-theoretic perspective to prove a strong version of the regularity lemma, which turns out to be extremely useful in his proof of the regularity lemma for hypergraphs. Tao's proof is, by far, the most concise proof for the hypergraph regularity lemma.
Let me try to explain at a very high level this information-theoretic perspective.
Suppose you have a bipartite graph $G$, with the two vertex sets $V_1$ and $V_2$ and the edge set E a subset of $V_1 \times V_2$. The edge density of $G$ is $\rho = |E|/|V_1||V_2|$. We say $G$ is $\epsilon$-regular if for all $U_1 \subseteq V_1$ and $U_2 \subseteq V_2$, the edge density of the subgraph induced by $U_1$ and $U_2$ is $\rho \pm \epsilon |U_1||U_2|/|V_1||V_2|$.
Now, consider selecting a vertex $x_1$ from $V_1$ and a vertex $x_2$ from $V_2$, independently and uniformly at random. If $\epsilon$ is small and $U_1, U_2$ are large, we can interpret $\epsilon$-regularity of $G$ as saying that conditioning $x_1$ to be in $U_1$ and $x_2$ to be in $U_2$ does not affect much the probability that $(x_1,x_2)$ forms an edge in $G$. In other words, even after we are given the information that $x_1$ is in $U_1$ and $x_2$ is in $U_2$, we haven't acquired much information about whether $(x_1,x_2)$ is an edge or not.
The Szemeredi regularity lemma (informally) guarantees that for any graph, one can find a partition of $V_1$ and a partition of $V_2$ into subsets of constant density such that for most such pairs of subsets $U_1 \subset V_1, U_2 \subset V_2$, the induced subgraph on $U_1 \times U_2$ is $\epsilon$-regular. Making the above interpretation, given any two high-entropy variables $x_1$ and $x_2$, and given any event $E(x_1,x_2)$, it is possible to find low-entropy variables $U_1(x_1)$ and $U_2(x_2)$ -- "low-entropy" because the subsets $U_1$ and $U_2$ are of constant density -- such that $E$ is approximately independent of $x_1 | U_1$ and $x_2 | U_2$, or that the mutual information between the variables is very small. Tao actually formulates a much stronger version of the regularity lemma using this setup. For example, he doesn't require that $x_1$ and $x_2$ be independent variables (though there hasn't yet been an application of this generalization, as far as I know).