This article may contain spoilers.
This dilemma can occur at any time in the library building, but has a higher chance of occuring based on certain survivor stats (most likely perception based).
When this dilemma shows up, the player will be shown a picture of a book case and given a description on the left side informing them of who found it and what it is. After this text, if you have a librarian in your group, it will display a message stating that the librarian in the group knows a secret passage in. At the very bottom of the left side, your chances of living through a 'break down the door' selection is displayed with 'those zombies will wipe us out if we barge in.' being the lowest possible chance of success.
On the far right side the following choices are displayed:
Leave it be.
Break down the door.
Let (Librarian survivor name) sneak in. (Only available if a librarian is present.)
Picking the first option will cause nothing to happen. (It is unknown whether you can get the dilemma to pop up again in the same playthrough or not.)
Breaking down the door will release a very large number of zombies for your survivors to fight off. (50+ usually)
Allowing your librarian to sneak in will usually result in them getting in and out without a problem. (It may be possible for another one of the survivors to get the librarian killed. Sometimes survivors will 'make noise' by accident, but this has yet to result in a dead librarian.)
If you break down the door and at least one survivor is alive after the zombies or if your librarian sneaks in, your group will receive information or a permanent stat boost to a specific task. (or other possible boosts.) It's possible that you will simply receive information about a single building in the city and what it contains, but it is also possible to receive a permanent boost to a task such a cooking food or barricading.