From my knowledge, the column space is the vector space that is spanned by the column vectors of a matrix. From my lectures, I've been told to find the basis of a column space of a matrix (unless I've forgotten) by analog to finding the basis for the row space of a matrix, which would be to reduce the matrix $A^T$ to row echelon form and note which rows have non-zero entries, just as similarly you would do for the basis of the row space - convert $A$ to row echelon form and take the rows that are non-zero.
But apparently, and which I find easier, the column space basis can be found by merely noting which columns in $rref(A)$ have a pivot, and then taking the original columns of $A$ as the basis for the column space. This doesn't make much intuitive sense to me. Why is this the case?