This seems simple but has stumped me for around half an hour...
The paragraph reads: "Let $x_1,x_2,x_3$ be three variables. We let the permutations in $S_3$ move around like $1$, $2$, $3$". So for instance, the permutation $(132)$ sends $$ \begin{align*} x_1 &\to x_3, \\ x_2 &\to x_1, \\ x_3 &\to x_2.\\ \end{align*} $$
I do not understand what this notation does to explain the permutation $(132)$ because surely $x_1 \to x_1$.
Sorry if this is stupid but I really cannot see it!
On
The symbol $(132)$ stands for the cyclic permutation that permutates the elements $x_1$, $x_2$, $x_3$ as follows: $x_1\mapsto x_3$, $x_3\mapsto x_2$ and $x_2 \mapsto x_1$.
Another example in $S_3$: the symbol $(13)$ stands for the permutation that interchanges $x_1$ and $x_3$, but keeps the element $x_2$ fixed.
On this wikipedia page on cyclic permutations there are some other helpful examples.
I think you are misreading the notation $(1 3 2)$. As Ernie060’s answer states, this represents the cyclic permutation $\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}$ where $1\mapsto 3$, $2\mapsto1$, and $3\mapsto2$. But based on the last part of your question, it seems like you are reading $(1 3 2)$ as the bottom row of the permutation $\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}$ where $1\mapsto 1$, $2\mapsto 3$ and $3\mapsto 2$. This is simply not what the author of the book means.
This is a very common and useful way to represent permutations, arguably more common than the functional notation. It would be reasonable to assume this as the default meaning if it isn’t otherwise specified (but it probably was specified).