How do I convert a summation equation to a vector equation (backpropagation)?
$a_j^l$ is the $j$-th component of $a^l$. Hence, the first equation is $1$ dimensional, the second equation is a vector equation. 

Similarly for notaiton $b_j^l$ and $w_{jk}$.

$\sigma$ means sigmoid function. If the input is a vector, then it evaluated the vector componentwise.

To answer your main quesiton, recall that from definition of  matrix vector product, where $A$ is a matrix and $x$ is a vector, $Ax$ produces a vector $y$ where the $j$-th component is $y_j= \sum_k A_{jk}x_k$.  

That is element of $$y=Ax$$

can be written as $$y_j = \sum_k A_{jk}x_k$$

elementwise.

Hence that is how we can "remove" the summation sign.