In the cited section of Conway an injective holomorphic function $h\colon G\to D$ has been constructed, where $G\subset\mathbb{C}$ is a domain. For a point $a\in G$, one then defines a function $g\colon G\to D$ by $ g(z) = \frac{|h'(a)|}{h'(a)}\frac{h(z) - h(a)}{1 - \overline{h(a)}h(z)}.$ My understanding of the question is: why is it that $g$ maps into $D$?
The answer to this question is that the function $\varphi(w) := \frac{|h'(a)|}{h'(a)}\frac{w - h(a)}{1 - \overline{h(a)}w}$ is an automorphism of $D$ (see p. 131 in Conway). The function $g$ is the composition $\varphi\circ h$. Since $h\colon G\to D$ and $\varphi\colon D\to D$, the composition $g$ is from $G\to D$.
Also, it would be more helpful for us and for future users with the same question if you wrote the question explicitly instead of providing a (fickle) link. Hope this helps.