Essentially, the Classical Latin now or until recently learned in schools faded out as a ‘birth language’ by no later than AD 100 – and for the last 100-150 years of its existence it was of the character of “U” British English – a ‘proper’ variant on the common speech preserved by a self-styled elite and those who looked to them for cultural leadership.
Alongside it, though, was growing Vulgar Latin, a colorful vivid mostly-oral language that is poorly documented, for the simple reason that what was written and preserved tended to be in Classical Latin. There were vocabulary differences, such as caballus for equus (horse) but major distinctions were in grammatical forms – the specialized -s forms of third-declension nominatives were largely discarded, the -us endings of second conjugation nominative transformed to the -o alreayd present in dative and ablative forms, verb conjugations simplified to a certain degree, and generally hints of what would someday become Romance languages emerged.
Somewhere around or after 400 AD this transmuted to Late Latin, only slightly better documented than Vulgar. But, more importantly, this development coincided with the crumbling of the Empire in the West. (Already a renascent Greek had asserted dominance over the East; Koine Greek had always been a lingua franca, even while the Empire was united and strong, and Latin could not compete with it in the East, except for Dacia.
The ‘Carolingian Renaissance’ brought new life to Classical Latin as the language of culture and such scence and scholarship as existed, but the popular forms were rapidly growing apart as dialects. Imagine, if you will, the fall of civilization in about 1900. Then contemplate reconnection, 300 years later, of Australia, Yorkshire, southern England, English-speaking Ireland, Canada, the Northeast U.S., the American South, California, and New Zealand, each with their own dialects, with a common trait of doing scholarly and cultural work in Victorian English while evryday conversation and trade is conducted in the local dialect. Such a picture gives you the beginnings of the Romance languages.
Eventually national languages based on the dialects of Paris, Toledo, Florence, Lisbon, etc., came to be regarded as ‘separate languages’ but local dialects persisted, all derived from Late Latin and some varying so far from the ‘national’ norm as to be mutually unintelligible. Sardinia, Provence, much of Spain, the Rhaetian valleys, Friulia, are among these – and the consolidation into ‘national’ standards is still far from complete.