Like Yoda says, "There is no why." There is no rhyme or reason to it. Mathematics is laced with a hodge-podge accumulation of inconsistent conventions that often make learning and doing higher math as cumbersome as learning and doing arithmentic with Roman numerals!
Somewhere along the line, someone defined matrix multiplication as, $C(i,j)=\sum_{k}A(i,k)B(k,j)$, when it could just as well have been defined as
$C(i,j)=\sum_{k}A(i,k)B(j,k)$,
$C(i,j)=\sum_{k}A(k,i)B(k,j)$,
$C(i,j)=\sum_{k}A(k,i)B(j,k)$,
$C(i,j)=\sum_{k}A(j,k)B(k,i)$,
$C(i,j)=\sum_{k}A(j,k)B(i,k)$,
$C(i,j)=\sum_{k}A(k,j)B(k,i)$, or
$C(i,j)=\sum_{k}A(k,j)B(i,k)$.
Someone else decided that the the composition of $a$ and $b$ should be written as $a*b$. Someone else decided that this was inconsistent and wrote the composition of $a$ and $b$ as $b*a$, when he could just as well have redefined matrix multiplication!
Conventions are not written in stone. If you don't like them, make up your own. That's what I do. For the composition of two permutations $a$ and $b$, expressed as row vectors in APL (A Programming Language), I write $c\leftarrow a$[$b$] and APL assigns the result of the composition to the variable $c$. I just use the same notation, replacing the $\leftarrow$ with an =. It turns out that the ] is actually redundant, so $a$[$b$ is just a rewrite of $a$*$b$.
If $A$, $B$, and $C$ are the corresponding matrices that represent these permutations, I write $C$=$AB$, where $AB$ is just the standard multiplication of $A$ and $B$. Problem solved.
How about an example? Consider two permutations in $S_4$, 2431 and 3214:
2431[3214]=3421
3214[2431]=2413
Let's see what happens when we multiply the corresponding matrices from the irreducible representation of $S_4$. These matrices permute the vertices of a tetrahedron at the $(x,y,z)$ coordinates $(6,4,3)$, $(-6,4,3)$, $(0,-8,3)$, and $(0,0,-9)$ among themselves. Making a matrix with these coordinate as the columns, we get:
$ \begin{array}{rrrr} 6 & -6 & 0 & 0\\ 4 & 4 & -8 & 0\\ 3 & 3 & 3 & -9 & \\ & \\ & \\ & \\ & \end{array} $
Multiplying this matrix from the left by the representation matrix for 2431, we get:
$ \begin{array}{rrrlrrrrlrrrr} -1/2 & -1/4 & -2/3 & & 6 & -6 & 0 & 0 & & -6 & 0 & 0 & 6\\ 1/3 & 5/6 & -4/9 & \times & 4 & 4 & -8 & 0 & = & 4 &0 & 4 & -8\\ 1 & -1/2 & -1/3 & & 3 & 3 & 3 & -9 & & 3 & -9 & 3 & 3\\ & \\ & \\ & \\ & \\ & \end{array} $
Multiplying the same matrix from the left by the representation matrix for 3214, we get:
$ \begin{array}{rrrlrrrrlrrrr} 1/2 & -3/4 & 0 & & 6 & -6 & 0 & 0 & & 0 & -6 & 6 & 0\\ -1 & -1/2 & 0 & \times & 4 & 4 & -8 & 0 & = & -8 & 4 & 4 & 0\\ 0 & 0 & 1 & & 3 & 3 & 3 & -9 & & 3 & 3 & 3 & -9\\ & \\ & \\ & \\ & \\ & \end{array} $
Multipying the representation matrices in the order 2431 $\times $ 3214, we get:
$ \begin{array}{rrrlrrrlrrr} -1/2 & -1/4 & -2/3 & & 1/2 & -3/4 & 0 & & 0 & 1/2 & -2/3 \\ 1/3 & 5/6 & -4/9 & \times & -1 & -1/2 & 0 & & = -2/3 & -2/3 & -4/9 \\ 1 & -1/2 & -1/3 & & 0 & 0 & 0 & & 1 & -1/2 & -1/3 \\ & \\ & \\ & \\ & \\ & \end{array} $
...which is the represntation matrix for the permutation 3421:
$ \begin{array}{rrrlrrrrlrrrr} 0 & -1/2 & -2/3 & & 6 & -6 & 0 & 0 & & 0 & 0 & -6 & 6\\ -2/3 & -2/3 & -4/9 & \times & 4 & 4 & -8 & 0 & = & -8 & 0 & 4 & 4\\ 1 & -1/2 & -1/3 & & 3 & 3 & 3 & -9 & & 3 & -9 & 3 & 3\\ & \\ & \\ & \\ & \\ & \end{array} $
...just as we wanted. You can check for yourself that multiplying the matrices in the opposite order gives you a matrix for the representation of the permutation 2413, as given at the beginning of this example - unless I have made some typing error here; this is VERY tedious!