Actually you need $ \mathbb{E}(Y \mid X=x) = \mathbb{E}(Y) + \frac{r\sqrt{\operatorname{var}(Y)}}{\sqrt{\operatorname{var}(X)}}(X - \mathbb{E}(X)). $
So $ \operatorname{var}(\mathbb{E}(Y\mid X)) = \left(\frac{r\sqrt{\operatorname{var}(Y)}}{\sqrt{\operatorname{var}(X)}}\right)^2 \operatorname{var}(X-\mathbb{X}) = \frac{r^2\operatorname{var}(Y)}{\operatorname{var}(X)} \operatorname{var}(X) = r^2\operatorname{var}(Y). $
In other words, the square of the correlation is what fraction of the variability of $Y$ is "explained" by the variability of $X$.
The "law of total variance" partitions $\operatorname{var}(Y)$ into "explained" and "unexplained" parts: $ \operatorname{var}(Y) = \operatorname{var}(\mathbb{E}(Y\mid X)) + \mathbb{E}(\operatorname{var}(Y\mid X)). $ In the special case you're considering, the second term---the "unexplained" part---is the expected value of a constant $\operatorname{var}(Y\mid X)=(1-r^2)\operatorname{var}(Y)$.