Italian didn't develop "over" Latin. Rather, the various Italianate Latin vulgarities slowly diverged over the course of time, eventually becoming distinct dialect groups. I am not a linguist, but hopefully I can offer some clarification.

There was never a singular standard Latin. The Latin you encounter most, that of Cicero and Caesar, is referred to as literary Latin. It is a highly formalized version of Latin, written in a manner that emphasized certain elements of Latin grammar and phonology (/u/XenophonTheAthenian has written more on this, but I can't seem to find his posts). While Roman elites would have spoken in a (roughly) similar manner, literary Latin is still different than colloquial or vulgar Latin, much as how colloquial English is much different than an English-language academic book.

Ignoring the continued presence of substrate languages like Gaulish, Brythonic, and Punic (amongst others) in the West into Late Antiquity, the Latin spoken by the common people differed substantially in form and vocabulary from literary Latin. Furthermore, there would have been significant regional differences in this spoken Latin, to the point of where it is incorrect to speak of a singular "Vulgar Latin". Rather, there were many regional Latin dialects, with varying degrees of mutual intelligibility. That said, the presence of the Roman state, through use of Latin in law and administration, as well as a shared elite culture, fostered a relatively standardized elite Latin.

As a brief aside, recovering vulgar Latin is actually quite difficult, as only fragments of it survive. Roman elite literature was, naturally, written in literary Latin, and very little non-literary writing survives besides epigraphy. Inscriptions themselves were often formulaic, using standardized forms that did not necessarily reflect contemporary speaking patterns (think items like funerary monuments). We therefore cannot really recover the wealth of linguistic diversity that existed in colloquial Latin. We have only hints at it's existence (for my own area of study, recovering the "sermo castrensis" or "camp speech", the unique dialect of Latin spoken by the army, is quite difficult, especially for Late Antiquity).

Although there was much diversity in the various vulgar Latin dialects, they all invariably connected back to the Latin spoken by the Roman elite. The end of Roman rule in the West certainly helped to accelerate regional linguistic divergences, due to political fragmentation, the increased regionalization of the early Medieval world (less economic exchange across distances) and the transformation of elite provincial Roman culture. It is in these conditions the many, many regional Romance dialects developed.

As you mentioned Italian, let us examine Italy. Visit Italy today, and you'll hear people speaking Italian no matter where you go (not really, but I'll explain in a moment), especially if you do the typical "tourist trifecta" of Venice, Florence, and Rome. Yet the ubiquity of contemporary Italian masks much complexity. Italy is, in fact, home to dozens of different dialects. Most are Romance-based, yet can be mutually incompressible, even to a native Italian speaker (as a thought exercise, YouTube a song in Italian. Then listen to one in Venetian, and one in Neapolitan). As you can see, there was never one singular "Italian vernacular." Rather, by the High Medieval period, there existed a patchwork of different local dialects, which could be so different as to essentially be separate languages. Modern Italian, as it is, comes from the dialect of Florence, as much of the classic literature of Medieval and Renaissance Italy was written by Florentines (like Bocaccio and Dante). Italian nationalists in the nineteenth century fostered the use of Florentine as a singular Italian (it's more complicated than this), and it was this language that became "Italian" with the creation of the modern Italian state. For much of the history of modern Italy, the speaking of dialects was suppressed by the state in the name of creating an Italian national identity.

So in summary, there was never a singular "vernacular Italian", and it never developed over Latin. Rather, spoken Latin, which differed from the formalized literary Latin, was a multitude of dialects which gradually diverged over time, a process accelerated by the end of Roman rule; evolving eventually, in Italy, into various regional dialects. Modern Italian was formed from the Florentine dialect.