It's just linear algebra. Let's take a concrete example where $n=3$. Then ${\bf \mu} \in \mathbb{R}^{3 \times 1}$. 

$$\Sigma_{MLE} = \dfrac{1}{3}\sum_{i=1}^3(x_i-{\bf \mu})(x_i-{\bf \mu})^T = \frac13 \left( \begin{array}{c} x_{1} - \mu \\ x_{2} - \mu \\ x_{3} - \mu\end{array} \right)\left( \begin{array}{ccc} x_{1} - \mu & x_{2} - \mu & x_{3} - \mu\end{array} \right) $$
$$ = \frac13  \left( \begin{array}{ccc} (x_{1} - \mu)^2 & (x_{1} - \mu)(x_{2} - \mu) & (x_{1} - \mu)(x_{3} - \mu) \\  (x_{2} - \mu)(x_{1} - \mu) & (x_{2} - \mu)^2 & (x_{2} - \mu)(x_{3} - \mu) \\ (x_{3} - \mu)(x_{1} - \mu) & (x_{3} - \mu)(x_{2} - \mu) & (x_{3} - \mu)^2 \end{array} \right) $$
$$ = \left( \begin{array}{ccc} Var(x_1) & Cov(x_1,x_2) & Cov(x_1,x_3) \\  Cov(x_2,x_1) & Var(x_2) & Cov(x_2,x_3)\\ Cov(x_3,x_1) & Cov(x_3,x_2) & Var(x_3) \end{array} \right) $$

Also note that $Var(\sigma_{11})$ is odd notation. You might be thinking $\sigma^2_{x_1}$ or $Var(x_1)$.