After six centuries of remaining locked into one specific form, it was suddenly allowed to evolve again.
On its own, spoken and written language don't just stay the same for hundreds or thousands of years: it changes as people use it. New words are invented for new ideas and concepts, old words have their definitions and spellings changed and updated. Some words fall out of use and become archaic or forgotten. We see that today, in our increasingly global, always-connected 'online' culture, with so many new words being created and old words being redefined and repurposed, new spellings and increasingly fluid use of language at all levels of society.
During the high Imperial period of Roman civilization, written Latin essentially froze in place, departing from this normal cycle of change and evolution. Even as Roman Imperial civilization grew and expanded and matured and eventually aged into its 5th century decline, a very formal, archaic style of Latin (of the sort which had been employed by Caesar, Cicero, Horace, Ovid, Livy, and other prominent writers of the so-called 'Golden Age' of Latin literature) which had been in use during the Caesarian/Augustan era (1st Century BCE) remained in use both among the wealthy and well-educated elite and the Roman Imperial army and bureaucracy. The language froze in place for nearly six centuries. There are numerous theories as to why this might have been the case (the old, wealthy and well-read Senatorial elite who produced the bulk of our surviving Latin sources today may have held onto the written language of the last epoch of Roman history in which Senatorial status had really counted for something real in terms of political power, or perhaps the quickly Romanizing Western Empire latched on so tightly to the works produced in the 'Golden Age' of Latin literature and sought to reproduce that style of writing, in order to 'prove' their 'Romanness' that the written, formal language went into a sort of feedback loop--but as far as I know no one has produced a sufficiently compelling single argument for the cause as to win over the majority of Classicists.) Though, of course, behind the scenes commonly-spoken Latin--'vulgar' Latin is the technical term--was continuing to evolve, its evolution was checked by the basic need for people from different parts of the Western Empire (such as Hispania, Britannia, Italia, North Africa, Illyricum, and Gaul) to continue to be able to communicate with one another, and in communication with the 'frozen' state of the language which was being used in formal, Imperial contexts, for things such as Imperial proclamations, military and civic correspondence, legal judgements, inscriptions, and so forth. For six centuries or so, the children of middle-class and elite families had this essentially-already 'dead' form of Latin beaten into their heads by their teachers and tutors, and 'common' military enlistees who joined the Legions (especially Centurions who rose through the ranks) had a rudimentary form of it pounded into them by the intensely bureaucratic nature of Roman army life and logistics. They would then take that archaic form of Latin back home with them when they left the service, and communicate it to their wives and children and neighbors. This effectively kept both spoken and written Latin relatively stuck in the same sort of shape for six centuries all across the Western Empire.
When the Western Empire collapsed, however, in the late fifth century, the various individual regions of the former Imperial civilization which had all been speaking and communicating in Latin with one another for the past six hundred years suddenly began going their separate ways, not only politically, culturally, and economically, but linguistically as well. In those former provinces were not conquered outright by non-Latin speaking foreign peoples (such as Britain by the Angles and Saxons in the 5th and 6th centuries, North Africa by Islamic invaders in the 8th, Illyricum and the Danube Provinces by Slavic and Turkic invaders from the 7th century onwards), Latin remained the dominant language, but with a twist. Formal education (both of the elite and non-elite sorts--elite education continued in only a drastically limited sense under the auspices of Church officials) being one of the casualties of the dissolution of the broadly-integrated pan-imperial civilization, the gradual evolution of the spoken language (which, remember, had been going on all along, merely kept in check by the need to make sure that the various disparate regions of the Western Empire could all communicate with one another and with the formal Latin being spoken and written by the central Imperial government that held the system together) was suddenly unleashed to run rampant throughout society. Though it would take centuries more before political and economic and linguistic consolidation would produce the four principal organically produced Romance languages (in Gaul, what we call French today after the Frankish overlords who came to rule the former province evolved in one way, where as in Eastern Hispania, Spanish evolved another way, in Western Hispania Portuguese, and in Italy, Italian went its own way) and the many disparate regional dialects which still survive today in small enclaves all across the map of Western Europe (including Sardinian, Occitan, Catalan and so forth.) Of these, as you might imagine from the fact that Italy had been both the seat of Empire for most of Imperial History as well as the home of the Papacy in the Medieval Period, Italian and Sardinian remain the closest to the old Imperial Latin, but all have diverged significantly.
The written and spoken language of the Church, however, remained Latin, hewing closely to that old 'Classical' tradition of essentially FROZEN Latin which was not allowed to change or naturally develop. Although 'Church Latin' WOULD evolve somewhat during the Medieval Period, as Church Scholars would continue to write new theological treatises and produce new liturgical compositions down through the ages and imperfectly transmit the Latin of the Augustan Era through the next thousand years, these evolutions were so minute in comparison to those which 'vulgar' Latin underwent in its various regional (and later regnal) dialects that anyone with an education in Classical Latin can easily read Medieval Church Latin, and vice versa.