There isn't a clear consensus and the historical records of that time(the entire first millennium) because we do not have written documents from the Romanians and the mentions by neighboring states(Byzantine Empire or the Bulgarian Empires) are few and very short.

That being said, there are two main schools of taught. The Romanian historiography states that the language and the people developed naturally from the mix of Dacians and Romans that remained here after the retreat of the Roman administration from Dacia. The Hungarian historiography says that the former province of Dacia was deserted(in an extreme cases it is considered completely deserted) and there was a latter migration of Romanian speaking population from the Balkans towards present day Romania. That romance population in the Balkans exists even today, aromanians being one of them, but the main problem is that we do not have written or archaeological evidence regarding such a migration. Those two different opinions have their trace in the XIX century nationalist movement.

Now, for one of the reasons Romanian survived here is because there was no assimilationist pressure from the populations that migrated in the region after the Romans left. Some migratory tribes just passed(the Mongols towards west). Some, like the Avars, Cumans or the Gepids did not manage to survive as a ethnic group by themselves, let alone assimilate others. Finally, some like the Slavs had indeed a big influence on Romanians: we have lots of Slavic words in our language and old Slavonic was used by the Orthodox church. In Gesta Hungarorum, one of the 3 leaders mentioned when the Hungarian came was a Vlach and a Slav.

One thing remains. When I spoke about the territory in my comment, I said the Province of Dacia, but Romanians today encompass a way larger area(like all Wallachia and Moldavia). In the case of Moldavia, we have indeed a migration. Regarding this event, there are records and we do know what happened. Basically, Moldavia was created as a buffer zone in the east by the Hungarian Kingdom. We know for certain that some Romanian population from north of Transylvania(Maramureș) crossed the Carpathians into Moldavia(in the XIV century). The first leader to do that was Dragoș. His son was deposed by Bogdan of Cuhea, also from Maramureș.