relation between Galois group and ramification type of polynomial over a function field

In chapter 4 of J. P. Serre's "Topics in Galois Theory", he computes the Galois groups of the splitting fields over $\mathbb Q(T)$ of a few polynomials of the form $f(X,T)=f(X)-T$. He does this by calculating their ramification type (i.e. which valuations ramify in this field extension with which ramification indices) and then from that drawing conclusions on elements of the Galois group. However, I fail to see the connection between the ramification type and the Galois group.

An example (p.42): The polynomial $$f(X,T)=X^n-X^{n-1}-T\in\mathbb Q(T)[X]$$ has (by straightforward computation) the ramification type $$\begin{cases}v_\infty:&\text{degree }n,\\v_{X}:&\text{degree }n-1,\\v_{X-\alpha}:&\text{degree }2\quad \text{(for }\alpha:=\frac{n-1}{n}\text{)}.\end{cases}$$ From this, he seems to draw the conclusion that $\operatorname{Gal}(f(X,T),\mathbb Q(T))$ contains an $(n-1)$-cycle and a transposition. I would like an explanation as to why this follows. The rest of the proof is clear to me.