Languages aren't usually made up. and if one were, it would be difficult for anyone to know fluently since fluency usually requires learning it as a child and/or immersion. Usually, new languages evolve from other languages when a group of people lives separate enough from another group that speaks the same language for long enough that the language changes (this happens faster for languages that do not have a written component).
The question though is why would they make up a language? Keep in mind that a language is more than a vocabulary, there is grammar as well, syntax, sentence structure, congugations, etc. All of that would be a lot to come up with, learn, remember and teach. What seems more likely is that it would have started with a secret code or short-hand based on one or more existing languages. (Think Yiddish which is a combination of German and Hebrew). Perhaps the religious sect was being persecuted by the mainstream society and had to come up with a code to communicate. Perhaps the persecution lasted long enough that children learned the secret code as their primary language. (again, think Yiddish) Perhaps the persecution is over now but the language that evolved from the secret code persisted. 634
Perhaps those of the religious order have been isolated for so long that what was once the same language everyone else speaks, has evolved into a full-blown different language. If this language has a new writing system, perhaps it started as a form of short-hand they learned for record keeping and evolved from there. Perhaps children are dedicated to the monastary at a very young age and grow up speaking this new language instead of that of their parents or perhaps the <PERSON> family is part of the religious order and therefore speaks the language.
The above would have taken generations though. If you want something faster, maybe the language didn't evolve. Maybe it was revealed by one of the gods and the minds of the few faithful monks were all at once enlightened to it so that the god can communicate with them without the traitors or unfaithful of his sect understanding. Maybe it is still to avoid persecution from the followers of a different god. Maybe it is so that god can make the other gods jealous. Or maybe it is for some reason he hasn't revealed yet. Another possibility is that they have discovered an ancient text written in a long forgotten language.
TL;DNR I guess my answer to the question "Why would monks make up a language?" is "They probably wouldn't." But there are other, more natural ways to have a group of monks speak a language than the society around them, such as a language evolving from a secret code or revelation of the gods. 209