The above page is from Complex Analysis by Stein.
In Theorem 3.1, we denote the function $f(z)$ holomorphic in the strip $S_b$ by $f_b$. When $0＜\alpha＜\beta＜a$, because $f_{\alpha}|_{\Bbb R}=f_{\beta}|_{\Bbb R}=f(x)$, therefore $f_{\beta}|_{S_{\alpha}}=f_{\alpha}$, we have $\bigcup_{0<b<a}f_b$ holomorphic in $S_a$, then $f(x)$ is the restriction to $\Bbb R$ of a function $f(z)$ holomorphic in the strip $S_a=\{z\in\Bbb C:|\operatorname{Im}(z)|<a\}$, isn't it?