Together with a mailbox message, an email comes to the registered email address (if the user registered by google, the SE can see his email address). The email restates the mod message, and it contains a link to the moderator message page. Thus, you can be sure that the user receives your message, except if he registered from a rarely visited fake mail account.
The URLs follow the common CRUD urls of the SE, like
The psychological pressure is the highest possible in this context, because it doesn't happen without cause. Practically, if a mod writes anything, it is possible that it is only a warning, but it is always about suspension.
The user has only a single answering possibility, and he is warned for that. If effectively means for him that what he says doesn't matter, it will be probably rejected. He also has many reasons to suspect that the possibility to "appeal" to the SE is only theoretical.
Thus, if he doesn't react, or reacts some impolitely, he knows that anything he does won't make his case better, and also not worsen it.
It is also possible that he thinks you are right, but he won't admit it.
Such communications are probably preserved until eternity and regularly re-visited in case of the possibility of a new suspension. Mods and higher entities can probably see all of them, simple users can only see theirs. But regular users can't see the links again after they were removed from their inbox (it is similar to how it is with deleted posts - theoretically you can see them, but if you didn't save their links, you can't find them).
In most cases, the mods try to communicate the cage decision in a relatively cooperative way, and I think many users can think that you also have to fulfill contradicting requirements. The treatment of the SE mods is much better than on most other sites. But these words are also threatening with suspension, and the user - particularly in the case of the longer ones - can consider them unjustified (related example).