If the true success probability for binomial data is close to 0.50, why would you expect to have less certainty with your mean parameter estimate than if the true success probability were closer to 0 or 1.
When $p=0.5$, each single experiment, say coin toss, has greater uncertainty than any other $p$. For example, if $p$ was $0$, all coin tosses would turn up Tails, and there'd be no uncertainty over the results. So, if a single experiment result is more uncertain for $p=0.5$ compared to other $p$, we'd also expect the mean of multiple experiments to be more uncertain. Here, I assumed the uncertainty is defined by the entropy (or the variance).