But if you click that button, well, they too can see what's forbidden:
The users who voted to close aren't meant to be kept secret. The panel that displays who voted to close (and whether or not they're a gold badge holder using their dupe-hammer powers) isn't useful to newer users who don't have access to close/reopen votes.
In essence, post notices were designed to be as succinct as possible, and only display relevant information to relevant people. The question author gets information on what to do next if their question is closed, and close voters get to see who voted to close and for what reason in this panel because they have an active interest in casting votes to reopen, or editing the question so that it's reopenable (if possible.)
Users who do not have access to this privilege do not have such access, and thus don't really need to see the expanded information, but it's not meant to be secret. You can browse the timeline or, as rene pointed out, browse the Data Explorer.