Is my interpretation correct?
Given a matrix $A$, you can form another matrix by taking only some of the rows and columns of $A$, possibly repeated. For example, suppose that $A$ is the following matrix:
$$
\begin{bmatrix}
1 &amp; 2 &amp; 3 \\
4 &amp; 5 &amp; 6 \\
7 &amp; 8 &amp; 9
\end{bmatrix}
$$
If we take only rows 1,2 and columns 1,3, then we get the matrix
$$
\begin{bmatrix}
1 &amp; 3 \\
4 &amp; 6
\end{bmatrix}
$$

For your second question, $\mathcal{D}_x$ is a distribution on column indices. You don't choose a column directly – you only choose an index. This corresponds to 1,3 in the example above.