It helps to think of bases first. Consider the Euclidean space $\mathbb{R}^{n}$, spanned by $\{x_{1},\dots,x_{n}\}$. The standard basis for this space is $\left(\begin{matrix}1\\ 0\\ \vdots\\ 0 \end{matrix}\right),\left(\begin{matrix}0\\ 1\\ \vdots\\ 0 \end{matrix}\right),\dots,\left(\begin{matrix}0\\ \vdots\\ 0\\ 1 \end{matrix}\right)$, but you can use any set of vectors (though using this standard basis makes it a little bit easier to understand what is happening). A matrix (or linear transformation $A$) provides the directions, so to speak, of how to map these vectors into your new space $\mathbb{R}^{m}$. Let $A=\left(A(x_{1})|\cdots|A(x_{n})\right)$. The column $A_{i}$ provides the vector that $x_{i}$ maps onto under the transformation of $A$. For now, don't think so much about the coefficients. Think more about the columns. For example the matrix $ A=\left(x_{1},0\dots,0\right) $
maps the $x_{1}$ onto $x_{1}$ and annihilates the other $x_{j}$. Now if we consider any vector, $X$, we can write it in terms of the basis vectors of $\mathbb{R}^{n}$, as $X=\sum_{i=1}^{n}c_{i}x_{i}$, and we can see what $A$ does to it. Simply take $AX=A\sum_{i=1}^{n}c_{i}x_{i}=\sum_{i=1}^{n}c_{i}Ax_{i}=\sum_{i=1}^{n}c_{i}A_{i}$ by linearity, where $A_{i}$ is the $i^{th}$ column of $A$. Thus we can see that $A$, in this case, extracts the first coordinate of $X$ and multiplies it by $x_{1}$. We can consider more complicated examples $ A=\left(x_{2},x_{1},0,\dots,0\right) $
In this example, $A$ sends the first basis vector to the second basis vector, and the second basis vector to the first basis vector. So we have $AX=c_{1}x_{2}+c_{2}x_{1}$.
Now we can consider matrices $A$ that are spanned by linear combinations of the $x_{1}$. For example $ A=\left(x_{1}+x_{2},0,\dots,0\right) $
sends $x_{1}$ to $x_{1}+x_{2}$. i.e. we would have $AX=c_{1}(x_{1}+x_{2})+0$. Many of the transformations we have can be thought of in the same way.
Consider rotation matrices. Rotation matrices send basis vectors $x_{1},\dots,x_{n}$ to a new set of basis vectors that have been rotated by some set of angles $\{\theta,\phi_{1},\dots,\phi_{n-1}\}$. Using what we have seen above, we can find what the matrix does to any vector $X=\sum_{i=1}^{n}c_{i}x_{i}$.
Hope this helps!