Construction of Riemann Sphere $C_{\infty}$ in Miranda's "Algebraic curves"

Question

I have a question about some steps in the construction of $C_{\infty}$ in an excerpt of Miranda's "Algebraic curves and Riemann surfaces" (page 61):

That's not clear to me how the map $\pi:Z \to C_{\infty}$ arises from the meromorphic function $x$ on $X$ and why does $\pi$ has degree $2$?