You must realise that a $n\times m$ matrix $A$ encodes a linear map $\Bbb R^m\to\Bbb R^n$ (note the order of $m,n$) and that the spaces at departure and arrival are otherwise unrelated. Moreover, matrices are usually just a tool to numerically represent a linear map $f:V\to W$ between two more abstract vector spaces, after choosing a basis in both; that actual matrix entries one gets depend heavily on the basis used. As a consequence there cannot be a direct geometrical relation between columns, which represent vectors of $W$ in coordinates with respect to the basis chosen there, and rows, which represent linear combinations of coordinate functions (for the chosen basis) on$~V$.
Nonetheless there is a way to understand what is going on from two points of view.
Solving a system of equation means trying to find the pre-image$~x$ (which must live in$~V$) under $f$ of a given vector $y\in W$.
The column picture says that the columns of $A$ describe particular vectors in$~W$ that are $f$-images of the chosen basis in$~V$; the question then is to find a linear combination of these that equals the given$~y$, and the corresponding linear combination of the basis vectors of $V$ will be our solution $x$ (of course there might be no solution, or infinitely many of them).
The row picture uses the coordinates in$~W$ with respect to the chosen basis, one at a time. In order for an $f$-image to match$~y$, each of its coordinates must be right. Computing one coordinate of the $f$-image of an unknown $x$ amounts to computing a linear combination of the coordinates of$~x$ with respect to the basis chosen in $V$, and the coefficients of such a linear combination are given by a row of $A$.
Requiring that one coordinate of the image matches the one of $y$ means that this linear combination of the coordinates of$~x$ must have a specific value, and that (usually) defines a plane in $V$. Combining these requirements means intersecting the planes in $V$ to find the set of possibilities for$~x$.
