A lot of times they are closed silently; they accumulate a pile of downvotes, eventually get 5 close votes, and disappear with no indication to the user of what has gone wrong.
This is not correct; there is a clear indication of what has gone wrong:
I think the author of the question gets an extended message with more information, but I am not sure where I can find its exact content (other than asking an off-topic question justI think the author of the question gets an extended message with more information, but I am not sure where I can find its exact content (other than asking an off-topic question just for this purpose). EDIT: thanks to Martin Sleziak for pointing out to me in chat that this purpose)is no longer the case.
If you believe that this closing message is not helpful enough, please feel free to suggest improvements. What you suggest here is that someone types a duplicate of that message in a comment on every off-topic question, which seems time-consuming and sub-optimal.
Also, keep in mind that new users have to click through another window that warns them that this website is for math research questions:
So the blame is at least partly on them for not reading the instructions. (We may also suggest improvements to these instructions.)