I am new to learning differential geometry, I am using the book "Problem Books in Mathematics" writing by Peter Winkler.
In the image below I don't know how he found the change of coordinates $ψ ◦ \phi^{-1}$ : I found that $\phi (U\cap V)=(0,\pi)$ so why he take $\alpha$ from $(\pi,2\pi)$
The image of the intersection $\phi(U \cap V)$ actually contains two pieces, corresponding to points with angle in the range $(0, \pi)$, and points with angle in the range $(\pi, 2\pi)$.
You can think of $U$ as the whole circle minus the point $(1,0)$, and $V$ as the whole circle minus the point $(-1, 0)$. So, the intersection $U \cap V$ is the whole circle minus both of these points.
Edit: Let's compute the transition maps. Starting with $\alpha \in (\pi , 2\pi) \subset \mathbb{R}$, $\phi^{-1}$ maps $\alpha$ to the point $p = (\cos(\alpha), \sin(\alpha))$. Then $\psi$ maps $p$ to its angle measured in the range $(-\pi, \pi)$. But our $\alpha$ is out of that range, so we shift it down by subtracting $2\pi$.