The proof in question, on MO, is as follows.
"Here's another proof, which shows that any connected paracompact locally Euclidean space X is second-countable. Cover X by Euclidean charts and take a locally finite refinement. Say an open set is good if it only intersects finitely many of the charts. Now take any point x and take a good neighborhood of it. The charts that intersect that good neighborhood can then themselves be covered by countably many good open sets. There are then only countably many charts intersecting those good open sets, and those charts can be covered by countably many good sets. Iterating this countably many times, you get an open set U associated to x which is covered by countably many charts such that if a chart intersects U, it is contained in U. It follows that the complement of U is also a union of charts, so by connectedness U is all of X. Thus X can be covered by countably many charts and is second-countable."
I do not know why we should expect this to be true after "iterating countably many times". It seems like this might follow from paracompactness, but I am not seeing how.
Best Answer
Let $U_0$ be the original good nbhd of $x$, and let $\mathscr{U}_0=\{U_0\}$. Let $\mathscr{C}_0$ be the collection of charts that meet $U_0$; $\mathscr{C}_0$ is finite, and each $C\in\mathscr{C}_0$ can be covered by a countable family $\mathscr{G}(C)$ of good open sets. Let $$\mathscr{U}_1=\mathscr{U}_0\cup\bigcup_{C\in\mathscr{C}_0}\mathscr{G}(C)\,.$$ this is a countable family of good open sets. Given a countable family $\mathscr{U}_n$ of good open sets, let $\mathscr{C}_n$ be the collection of charts that meet some $G\in\mathscr{U}_n$; $\mathscr{C}_n$ is countable, and each $C\in\mathscr{C}_n$ can be covered by a countable family $\mathscr{G}(C)$ of good open sets. Let
$$\mathscr{U}_{n+1}=\mathscr{U}_n\cup\bigcup_{C\in\mathscr{C}_n}\mathscr{G}(C)\,,$$
and continue, recursively constructing $\mathscr{U}_k$ for each $k\in\Bbb Z^+$.
Now let $U=\bigcup_{n\in\Bbb Z^+}\mathscr{U}_n$. Clearly $U$ is open. Suppose that some chart $C$ meets $U$. Then $C$ meets $\bigcup\mathscr{U}_n$ for some $n\in\Bbb Z^+$, so $C$ meets some $G\in\mathscr{U}_n$. But then $C\in\mathscr{C}_n$, so $\mathscr{G}(C)\subseteq\mathscr{U}_{n+1}$, and
$$C\subseteq\bigcup\mathscr{G}(C)\subseteq\bigcup\mathscr{U}_{n+1}\subseteq U\,.$$
This is a fairly common kind of construction of the closure of a set under some operation, in this case the operation of adding any chart that meets a set.