Languages are living things that exhibit changes all the time, often in unpredictable ways (even if stiff grammarians sometimes choose to ignore those changes). While changes can occur due to external influences, some language changes occur organically (can all changes to the English language between the time of Shakespeare and today be attributed to external influences or non-native speakers leaning the language?)

In keeping with this, Latin was not exempt from changes in its time as a "Living" language. 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 largely a convention born of convenience.

"Vernacular Latin" was being spoken long before the end of the Roman Empire, as ordinary people all over the empire developed regional patterns of speech and grammar. While vernacular forms of Latin were largely ignored by people leaving writing behind, this doesn't mean the language didn't very much exist for the innumerable people who used it in their everyday speech.

But Italians in positions to read and write (which would have been a small proportion of the population, at least compared to today) actually clung on to Latin for a very long time (certainly long after the end of the Empire) and kept using Latin for formal writing even after the vernacular diverged beyond mutual intelligibility and started causing problems (becoming a really noticeable force sometime in the eighth century, when lots of vernacular starts to noticeably creep into things like legal documents whenever the writer veers off-template). So to begin with, there is a pedantic debate to be had if examples of outstandingly "bad" or ungrammatical Medieval Latin (fundamentally, vernacular writing) is in fact the earliest form of Italian (for example, the "Placiti Cassinesi" or the "Veronese Riddle").

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 respective vernaculars for centuries: these are known today as the Dialects of Italy). I wrote this answer just a few days ago on the emergence, evolution, and adoption of "Standard" Italian, which you might be interested in.

As a final point, I would nonetheless point out that actual "Closeness" to Latin between Romance Languages is difficult to assess. Languages naturally exist along a "Continuum" with relation to each other, more so than they can be said to exist on any sort of timeline. So all Romance Languages are actually "closer" to each other than they are to Latin. And while predictably enough Modern Italian is still the "closest" Romance Language to Latin in terms of shared vocabulary and intelligible words, other languages (Spanish, for example) are closer to Latin in terms of shared grammar.

Thank you so very much! I really appreciate not only your knowledge, but also the time you have taken to share. Again, thank you!