The following is Theorem 16.15 in Rudin's RCA. After some searching, I figured out that there are a lot of different forms of the Monodromy theorem. So I wanted to attach the statement and proof of it.
Suppose $\Omega$ is a simply connected region, $(f, D)$ is a function element, $D \subseteq \Omega$, and $(f, D)$ can be analytically continued along every curve in $\Omega$ that starts at the center of $D$. Then there exists $g\in H(\Omega)$ such that $g = f$ on $D$.
((f, D) means $D$ is an open disc and $f$ is differentiable on $D$.)
Proof. Let $\Gamma_0$ and $\Gamma_1$ be two curves in $\Omega$ from the center of $\alpha$ of $D$ to some point $\beta \in \Omega$. Then the analytic continuations of $(f, D)$ along $\Gamma_0$ and $\Gamma_1$ lead to the same element $(g_\beta, D_\beta)$, where $D_\beta$ is a disc with center at $\beta$ (this is due to the theorem about analytic extension on a curve in simply connected domain). If $D_{\beta_1}$ intersects $D_\beta$, then $(g_{\beta_1}, D_{\beta_1})$ can be obtained by first continuing $(f, D)$ to $\beta$, then along the straight line from $\beta$ to $\beta_1$. This shows that $g_{\beta_1} = g_{\beta}$ in $D_{\beta_1} \cap D_{\beta}$. Then, $g(z):= g_{\beta}(z) (z\in D_{\beta})$ is well-defined, which is the desired extension of $f$$_\Box$
But I didn't understand the bolded part. Of course, if $f$ admits analytic extension along the sequence of discs $((f, D), \cdots, (g_{\beta_1}, D_{\beta_1}), (g_\beta, D_\beta))$, then, by definition, $g_{\beta_1} = g_\beta$ on $D_{\beta_1} \cap D_\beta$. But there may be a lot of sequence along which $f$ can be analytically extended.
All of these issues are tricky to express precisely without excessive verbosity. Rudin is perhaps a bit too concise here.
The first part that you understand asserts that the terminal germ $(g_{\beta},D_{\beta})$ obtained by path continuation along rival paths is independent of the path chosen. That succeeds in creating a well-defined germ of a solution to the global extension problem. Unfortunately, that terminal germ is only defined locally (on a small disk). We need to ask what happens if we change the center of the terminal germ? Does the germ at this new center (which we know exists) actually play well with the other germs we constructed (so that they all agree on overlapping patches? Rudin picks a nearby rival center $D_{\beta'}$ and shows that in this case of pairwise intersecting disks the answer is yes. Once this is shown, it should be easy to complete the proof by arguing that the maximal open region on which these local solutions patch together properly is the full path-connected region.