Let's say you're multiplying matrix $A$ by matrix $B$ and producing $C$:
$A \times B = C$
At a high level, $A$ represents a set of linear functions or transformations to apply, $B$ is a set of values for the input variables to run through the linear functions/transformations, and $C$ is the result of applying the linear functions/transformations to the input values.
For more details, see below:
(1) $A$ consists of rows, each of which corresponds to a linear function/transformation
For instance, let's say the first row in $A$ is $[a_1 \, a_2 \, a_3]$ where $a_1$, $a_2$ and $a_3$ are constants. Such row corresponds to the linear function/transformation:
$$f(x, y, z) = a_1x + a_2y + a_3z$$
(2) $B$ consists of columns, each of which corresponds to values for the input variables.
For instance, let's say the first column in $B$ consists of the values $[b_1 \, b_2 \, b_3]$ (transpose). Such column correspond to setting the input variable values as follows:
$$x=b_1 \\ y=b_2 \\ z=b_3$$
(3) $C$ consists of entries whereby an entry at row $i$ and column $j$ corresponds to applying the linear function/transformation represented in $A$'s $i$th row to the input variable values provided by $B$'s $j$th column.
Thus, the entry in the first row and first column of $C$ in the example discussed thus far, would equate to:
f(x=b_1, y=b_2, z=b_3) &= a_1x + a_2y + a_3z \\
&= a_1b_1 + a_2b_2 + a_3b_3