Latin didn't start to die out. Latin just became something that we gave other names to.

There are two different Latins you should be considering. One is spoken Latin, and the other is formal written Latin. The kind you might hear in a Catholic church service (Latin service is very rare but still happens in some places) is not how anyone ever spoke during the height of the Roman Empire. This is a modified pronunciation of the liturgical/formal/written language. I'm going to ignore that for this question, since it still exists but wasn't really the medium of chatting with your buddy Gaius down at the local mead shop.

The other Latin is often called Vulgar Latin, but this isn't a very good name. Can also be called colloquial Latin, spoken Latin, vernacular Latin. The main distinction is that this is how people actually spoke.

The question then is when did this die out, but the answer is that it didn't. The various dialects of vernacular Latin simply continued to diverge over time and as they became subject to other linguistic influences. Some of them became what you'd now call French, some Spanish, some Catalan, some Romanian.

These kinds of changes are happening in language all the time, including right now and with English. Your children's generation won't speak exactly like you do, just as you don't speak exactly like your parents. Language is constantly shifting, and if you pick any point in time as the start date and say "when did the language stop being like this point here", the answer is right away, or at least within a generation.

You could similarly ask when English started to die out. After all we aren't using an English anything like the English in Beowulf (to the average English speaker, at least). At what point did that stop being one language and start being another? There is no scientific answer to this, because it's all just a gradient of constantly shifting hues.

About a month ago someone asked the question What did Ancient Roman accents sound like, for which I gave an answer that you might be interested in.

tl;dr: Latin didn't die out. It just became something else, a process that started shortly after the first time someone spoke something you'd call "Latin".