See answers by /u/Daeres and /u/ConteCorvo and /u/bitparity to the thread When did the Romans become Italians?

More answers in the next comment.

I'll go ahead and offer my link-within a link tied to the answer that u/voyeur324 already indicated. I think this answer of mine is probably the best-written of the one's I've workled on to date.

Summarizing here, an important think to keep in mind is that Latin itself was not exempt from changes in its time as a "Living" language (as indeed all languages aren't). It is these changes which all eventually morphed into the individual romance languages. Indeed even over the span of its working life, Archaic Latin, Classical Latin, Late Latin, and Medieval Latin all exhibit distinct characteristics absorbed from a variety of external and internal influences. Where the line is drawn between varieties of Latin, as with varieties of any other language (and thus between Latin and its descendants), is really an arbitrary convention born of convenience. What is unmistakable though, it that ultimately over time these divergences did occur everywhere Latin was spoken. Italy is no exception, with countless regional vernaculars proliferating long before the end of the Roman Empire itself.

Once Italians grew comfortable with the idea that they did not always need to write in Latin, by a somewhat unpredictable sequence of events the poetic, literary, and philosophical works written in the Tuscan variety of the vernacular would become very popular among the literate class. As educated people all over the peninsula became familiar with Tuscan literary works, that language became the preferred standard for upper-class communication (most people, however, would continue to speak in their local vernaculars for centuries, which themselves were on a parallel tracks of divergence from Latin: these are known today as the Dialects of Italy).