Much of what is now Romania is congruous with the ancient Kingdom of Dacia, which was conquered by Rome in 106 CE under Emperor Trajan and held as the province of Dacia for ~250 years (although not entirely without conflict, revolution and reconquest). The Colonna Traiana, still standing in the Forum in Rome, celebrates Trajan's triumph in the Dacian Wars.

Here is a map of the Roman Empire at its pinnacle under Trajan in 117 CE, showing Dacia as a full province. Latin was of course introduced as the language of government and as with many other provinces in the Empire, rather quickly spread to the general populace (enough of it to entrench it as a common-use language, anyways).

As for why the populace continued to speak Latin (eventually evolving into Romanian) after repeated waves of non-Latin-speaking conquest and migration from the 4th century on: Questions like this are always a going to be tainted by speculation, and we know surprisingly little about the Dacians themselves, but it's likely that the area's relative geographical isolation played a part.

That's not to say that the Huns and especially the later Slavic migrants/invaders had no influence over spoken Romanian at various times, they most certainly did. Medieval and early modern Romanian sported a healthy crop of Slavic loanwords, and native pronunciation was also influenced by the Slavic languages. Part of why modern Romanian seems so much more conservative in comparison to the other Romance languages is that there was actually a 19th-century re-Latinization effort, scrubbing it of its Slavic elements, not unexpected given the widespread nationalistic trends throughout Europe during the 1800s.

I'll add as a simultaneously amusing and distressing aside that there is a lot of Romanian nationalist badling out there, which posits that Latin is actually a descendant of Romanian rather than the other way around. It's all part of a crackpot wag-the-dog theory based on the assertion that Romania is the cradle of all civilization.

Slavic culture just bypasses Romania... just kidding, the truth is that the Romanian language includes a lot of Slavic toponyms not mentioning common traditions (some pre-Christian), not as many as their neighbors but still. The Romanian language is Latin-based because there was always a dominant "majority" of Latin-based speakers in the area regardless of the migratory waves passing thru or settling in for a while until the next migratory wave replace them. Some left traces, others don't. Farming, shepherding or herding, mining, praying, honey and winemaking, trading and when circumstances require fighting, fleeing or hiding in the woods, looting, raping, killing, kidnapping, burning and of course impaling was centuries in the row the norm in the area.