Given a linear transformation $M:\mathbb R^n \to \mathbb R^m$, we can describe $M$ concisely by writing down the matrix $\mathbf M$ whose $i$th column is $M(e_i)$,
where $e_i$ is the $i$th standard basis vector. The matrix $\mathbf M$ could be called the matrix description of $M$ (with respect to the standard bases of $\mathbb R^m$ and $\mathbb R^n$).
Conversely, given an $m \times n$ matrix $\mathbf M$, there is a corresponding linear transformation $M$ which is described by $\mathbf M$.
If $v$ is an $n \times 1$ column vector, we can define the matrix-vector product $\mathbf M v$ by
\mathbf M v = M(v).
Note that if $v = v_1 e_1 + \cdots + v_n e_n$, then
M(v) &= M(v_1 e_1 + \cdots + v_n e_n) \\
&= v_1 M(e_1) + \cdots + v_n M(e_n) \\
&= v_1 \mathbf M_1 + \cdots + v_n \mathbf M_n.
So $\mathbf M v$ is a linear combination of the columns of $\mathbf M$,
which is a very useful way to think about matrix-vector multiplication.
If $\mathbf B$ and $\mathbf A$ are matrices, I think it's nice to define the matrix product $\mathbf B \mathbf A$ to be the matrix such that
(B \circ A)(v) = (\mathbf B \mathbf A)v
for all $v$.
The formula for the entries of $\mathbf B \mathbf A$ can then be discovered by choosing $v = e_i$, the $i$th standard basis vector.
The $i$th column of $\mathbf B \mathbf A$ is
(\mathbf B \mathbf A)_i &= (B \circ A)(e_i) \\
&= B(A(e_i)) \\
&= B(\mathbf A_i) \\
&= \mathbf B \mathbf A_i
where $\mathbf A_i$ is the $i$th column of $\mathbf A$. This
is the standard formula for matrix multiplication.