It didn't really. In Europe today there are close to 200 million people who speak Neo-Latin (aka Romance) languages, in addition to the well over half billion in the rest of the world. We just call them Spanish, French, Italian, Romanian, Occitan, etc.

The only reason that something called "Latin" isn't spoken today is that over a millenium and a half those who spoke Neo-Latin languages were divided geopgraphically and politically. Compare that with something like English, where what is called Old English and was spoken in the middle ages really isn't more similar to what we now now as English than Italian or Romanian is similar to the Latin of the late Roman Empire. The only real difference is that there's been an England but not one state that has been uniquely connected to Latin in that way, so we only have one English but many (Neo-)Latin languages.

Here's a previous answer by u/turboNOMAD that goes into more detail about how Latin worked in antiquity and how the breaking up of the Empire led to Latin developing into many different branches.