My thinking is that when all of the regions previously controlled by the Roman Empire separated, Latin evolved differently in each area because they were no longer united under the empire. The language kept differentiating until today where the romantic languages are all that's left...
The Roman state had a lot of bureaucratic jobs, get one of those and you were sorted for life. How to get one of these jobs? Be very well educated and have the rights contacts, and being good at Latin was a prerequisite and parents would pay handsomely for their kid's education.
The empire ceases, the jobs go, no reason to spend money on your kids education if there are no prospects for them. After the empire in western Europe, there were no jobs where being able to read or write well and thus no reason to learn how to do so. Any jobs which did require reading/writing were normally done by the clergy who of course kept Latin and literacy going through the medieval period. If I recall correctly, Peter Heather and Bryan Ward-Perkins have both talked a lot about this.
So yes, the fall of Rome rather ended Latin as a mainstream language. But at the same time, the Papacy in Rome did keep it alive.