The short version is "it's complicated!", but I'll try to give a slightly more detailed explanation here.
Rome established a single empire with a single language, currency and laws. This doesn't mean that other languages didn't survive - and even thrive - alongside the official Latin of the Romans. In the Eastern Empire particularly, we know that Greek and (albeit to a lesser degree) the Egyptian and Aramaic languages continued to be in use.
The pre-Roman languages of the Western Empire weren't written languages (neither were the languages of the 'barbarian invaders' that replaced Rome in the West), so we have less evidence of how well they survived alongside vulgar Latin. That they did continue in use through the Roman period, however, is shown by the survival of Welsh and Cornish for example.
The pattern of 'barbarian' incursions that caused the fall of the Western Roman empire, and created the new 'Barbarian kingdoms' of the West is complicated (to put it mildly!). Broadly, however, many of these 'barbarian' invaders weren't looking to create a new empire. They actually wanted to be more Roman. They occupied an area within the Western empire and proceeded to adopt its language and customs. In return, these 'barbarians' provided the stability and defence that the government in Rome (or Constantinople) no longer could. Over time, the languages of these 'Barbarian kingdoms' diverged, giving rise to the variety we see in modern Romance languages.
In addition to this, of course, Latin continued as the language of the Roman Catholic church. This also helped preserve the language in the West. In the absence of any alternative written language, Latin remained the main vehicle of communication for the learned classes. The influence of the Roman Catholic church is a large part of the reason for the influence of Latin on modern English.
Vulgar Latin also survived in the 'Barbarian kingdoms' which replaced the former Roman provinces of North Africa, right up to the Arab conquests of the late 7th century. After the conquests the policy of education in Arabic, and the restrictions imposed on non-Muslims meant that the usage of Vulgar Latin declined.
Britain, of course, is the big exception in the western empire. The fact that the Romano-British infrastructure and language declined so rapidly here suggests that the Romanisation of Britain was far from complete by the time that Honorius issued his famous rescript to the Civitates of Britain in 410AD.
The evidence for Dalmatia and Pannonia suggests that these areas suffered significant depopulation, probably due to Slavic invasions, from the 7th to 10th centuries. The invaders brought new languages which largely replaced the original Romance languages, although Dalmatian survived right up to the 19th century. Something broadly similar seems to have happened in other parts of modern Austria.