The answers already posted are quite nice, and do a good job of answering your general question of what a matrix inverse really represents. Let me look at your specific example in more depth, though, because the column-vector interpretation of matrices is sometimes useful.
Let's say you picked two points in the plane, $p = (5,-2)$ and $q = (7,3)$, and stuck them together as columns of a $2\times2$ matrix $A = \begin{bmatrix}5 & 7 \\ -2 & 3\end{bmatrix}$. What this matrix represents is the unique linear transformation that maps the unit vectors $\hat x = (1,0)$ and $\hat y = (0,1)$ to $p$ and $q$ respectively. (Try it out yourself: calculate $A\hat x$ and $A\hat y$ and see what happens.)
So what does $A^{-1} \approx \begin{bmatrix}0.103 & -0.241 \\ 0.069 & 0.172\end{bmatrix}$ represent? As others have said, it represents the transformation that undoes the transformation caused by $A$: it maps $p$ and $q$ back to $\hat x$ and $\hat y$. As it turns out, this is also the transformation that is undone by $A$. So another way of looking at it is that its columns represent the points, $r = (0.103, 0.069)$ and $s = (-0.241, 0.172)$, that $A$ maps to $\hat x$ and $\hat y$ respectively. (This is because, for example, $AA^{-1}\hat x = \hat x$, but $A^{-1}\hat x$ = $r$; plug that in and you get $Ar = \hat x$, or in other words, that $A$ maps $r$ to $\hat x$.)