It didn't die per se, any more than old english died, it just evolved. Much of the old Roman empire spoke some regional dialect of Latin, which would have been largely mutually intelligible. But as connections between those regions broke down after Rome fell each region's language diverged, resulting in the creation of the "romance languages" after centuries of change. Even so, some of the romance languages have incredibly strong similarities to one another and if you know, say, Italian or French it's a lot easier to pick up, say, Romanian.

Even so, Latin continued to be read and spoken continuously (through Christian religious institutions especially) up through the present day. But Latin stopped being used for day to day communication.

What makes a language dead or alive is basically whether or not it's taught to children and used daily for regular communication. Latin is today a dead language because only scholars, academics, and clergy learn it for study. For a long time Hebrew was also a dead language until Israelis started teaching it to their kids.

Languages don't have beginning or end points; there is no stage where Latin becomes French or Italian, other than an arbitrary one which we assign. In this sense, Latin never died; millions upon millions speak it today, but call it something different.

A more conservative form of the language persisted in church and written communication, but this was gradually taken over by the vernacular as the latter became more prestigious. This written standard was prone to interference from the vernacular, however, so wasn't actually fixed to any single point in Roman history.