For $\alpha \in \mathbb{R}^n$, let $s_\alpha$ be the reflection in $\alpha$, i.e. $s_\alpha$ sends $\alpha$ to $-\alpha$ and fixes pointwise the hyperplane orthogonal to $\alpha$. Then if $\alpha,\beta \in \mathbb{R}^n$ have an angle of $\theta$ between them, then $s_\alpha s_\beta$ is a rotation by an angle of $2 \theta$. Fine, I get that.
Here's what I don't get. Let $m$ be the order of $s_\alpha s_\beta$, where we see $s_\alpha s_\beta$ as an element of the orthogonal group $O(\mathbb{R}^n)$, and assume $m < \infty$. Then apparently (according to Humpfreys' Reflection Groups and Coxeter Groups) the angle between $\alpha$ and $\beta$ is $\pi - \dfrac{\pi}{m}$. Why is this the case?
If a rotation with angle $2θ$ has order $m$, then $2θm=2k\pi$, so that $θ=\frac km \pi$. Now why they choose $k=m-1$ has perhaps reasons that are outside these arithmetic considerations.
Let's reduce the situation to the most simple case, $α=e_1$, $β=ce_1+se_2$, $(c,s)=(\cosθ,\sinθ)$ so that the angle from $α$ to $β$ is $θ$ and anything interesting happens in the first two dimensions, with $$ S_α=\begin{bmatrix}-1&0\\0&1\end{bmatrix},\quad S_β=\begin{bmatrix}1-2c^2&-2cs\\-2cs&1-2s^2\end{bmatrix}. $$ Then their product is $$ S_αS_β=\begin{bmatrix}2c^2-1&2cs\\-2cs&1-2s^2\end{bmatrix} =\begin{bmatrix}\cos2θ&\sin2θ\\-\sin2θ&\cos2θ\end{bmatrix} $$ which is a rotation by $-2θ$. The smallest rotation in the positive sense can thus be achieved by the angles $-\frac1m\pi$ and $\pi-\frac1m\pi$ from $α$ to $β$.