Proposition 4.7(a) in Artin's book states:
Let $p,q$ be two permutations, with associated permutation matrices $P$, $Q$. Then the matrix associated to the permutation $pq$ is the product $PQ$.
The proof he gives is then:
Since $P$ operates by permuting rows according to $p$ and $q$ operates by permuting entries according to $q$, the associative law for matrix multiplication tells us that $PQ$ permutes according to $pq$: $$(PQ)X = P(QX).$$ Thus $PQ$ is the permutation matrix associated to $pq$.
I do not understand how associativity comes into play. Is the only idea that I want to apply the matrix $PQ$ (the product) to $X$, but this is exactly the same thing as first applying $Q$ and then applying $P$? Do I need to use anywhere that a composition of permutations is a permutation? In essence, that seems to be what we're using, so this is in effect a homomorphism between $S_n$ and the set of permutation matrices, if I am not mistaken.
You are right about the role of associativity; the idea is: lets write $X = \begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}$ with rows $x_1^t , \ldots , x_n^t$. Then $QX = \begin{pmatrix}x_{q(1)}\\\vdots\\x_{q(n)}\end{pmatrix}$ and therefore $$(PQ)X=P(QX) = \begin{pmatrix}x_{p(q(1))}\\\vdots\\x_{p(q(n))}\end{pmatrix}\overset{(*)}=\begin{pmatrix}x_{(pq)(1)}\\\vdots\\x_{(pq)(n)}\end{pmatrix},$$(In $(*)$ you need, that $p \circ q$ is again a permutation.) which shows that $PQ$ is the associated permutation matrix to the permutation $pq$.
You are right that this is a homomorphism $S_n \to GL_n(\mathbb R)$ which is probably what should be constructed here, as it is a representation of $S_n$.