This is one of those fascinating questions of history because the short answer is: we don't know. The long answer is: we really don't know. Let me explain.

Let's go back a bit and trace who these Romanians are claiming to be. In the early 2nd Century, the Emperor Trajan conquered the Dacians, a kingdom in modern day Translyvania. Over the following century, Roman power gradually expanded north of the Danube to include most of modern day Romania, included in the two main provinces of Moesia Inferior and Dacia. These areas along the Danube river roughly correspond to the limit of Greek cultural influence, and represent the eastern edge of Latin cultural influence; settlers to Dacia were mostly Latin speakers. Fast forward to the late 3rd Century and we have the emperor Aurelian abandoning the province thanks to crises at home. And here is where the narrative splits. Did the Latin speaking settlers of Dacia and norther Moesia abandon their homes and move south? Or did they stay, and refuse evacuation to safer pastures? There is some evidence that certainly some settlers did remain, as they did in Britain, and Romanian historiography suggests that these formed the core of the future Romanian people. However, there is little evidence that those settlements that continued to survive and remain occupied and survived (culturally at least) more than a century. Some Romanians would counter that these people moved to the mountains to literal safer pastures, and the fact that the Carpathians could act as a sort of bulwark of cultural survival does have some merit. Basically, the theory goes that these Daco-Romans abandoned their cities and villas, and moved to the mountains, where they holed up for the next thousand years. Now, if all this sounds fantastic, it's because it does border on it. However, it is certainly possible, so putting the theory to bed isn't entirely feasible. That said, it has plenty of problems.

For comparison, let's look to Britain, where Londinium was abandoned within a century into little more than a walled relic, and no remnants outside of a few scattered place names exist of the Latin tongue in Britain, despite it having had more than three hundred years of Latin rule and settlement, more than twice that of Dacia. Of course, this is not evidence unto itself, just a little context. So why do historians outside Romania tend to favour a later arrival date? Because there just isn't much of the way in evidence to suggest they've been there longer than about a thousand years. The lack of place names, a seemingly trivial issue to a lay person, is of particular importance. Almost all of the place names in Romania do not have Romanian origins, but rather are Romanianizations of German, Slavic, or even earlier names. This goes from towns to rivers to mountains. There's also the issue of the time period here. I mentioned Roman Britain above; is it really that plausible that a province that was settled for a mere century and a half could lead to more than 1700 years of cultural continuity in the area? Most historians would argue that is a bit too far fetched.

The alternative is that they were more recent immigrants to the area. Roman/Latin settlement of the Balkans and Dalmatian coast was fairly heavy, and was settled for a lot longer before Roman control broke down in those areas. These areas were particularly hard hit in the chaos of the Late-Imperial/Migration period, but also offer natural bulwarks against expansion in the form of their mountainous terrain. Within the mountains of the Balkans, pockets of culture could easily survive. And so, the theory goes, as the Balkans were invaded, first by Germans, and then by Slavs, over the course of the 6th to 10th century, the Romanized farmers of the plains were pushed to the mountains as their land was seized by the newcomers. However, these peoples survived in said mountains and carved out a pastoral lifestyle that gave them ample opportunity to spread along the mountain ranges of South-East Europe. This would be more in line with the place names origins in Romania being of Slavic or German origin, as if they were late comers who pushed off Slavic and German peoples from land as they moved down from the mountains it would make a lot more sense. And, if those peoples were from areas to the south of Romania, where Romanization had longer to take root and survived well past the fall of the Empire, it also makes a lot more sense. There is also a possibility that the reality is a mixture of these two. Did some Daco-Romans survive, and their survival led to the migration of other Romanized peoples to the area after the invasion of the Slavs? This is a distinct possibility as well. However, the best evidence (mostly determined by a lack of evidence) suggests a later date for Romanian arrival.

However, the former is still a popular theory in Romania. One can easily see why. If you are the cultural and ethnic descendents of peoples who have lived on the land for almost two thousand years, that gives you more claim than the recent newcomers, who have only been there for about 1300 years. When you consider the complicated cultural and territorial ambitions of this region, you might understand why that is important.

I'm not a credentialed linguist, but I have an interest in languages and lectured at three Romanian universities in 1997 on a U.S. Information Agency program. Just to buttress the historical account Spoonfeedme has provided, I would note that this particular "Romance" language is (I believe) more infiltrated by other language families than any other.

Yes, the third-person plural of "to be" is "sunt" -- exactly as in Latin. But the word for "yes" is "da," for heaven's sake! Russian and other Slavic influence is present in many nouns, as is Turkish and maybe even Semitic. (I have wondered, for instance, about the fact that the root "Tiner-" connotes youth, as in Bucharest's Parcul Tineretului -- Park of the Youths. Compare with the Hebrew word for "baby": "tinok.")

I would just like to point out that Romanian is not the only oddity in Europe in that sense. There's also Euskera in Basque Country, Hungarian, Albanian, Georgian and others. All these are languages either language isolates or surrounded by languages from other family branches

EDIT: On the same note, you could ask why don't they speak a slavic language in Hungary? Or a germanic language in Finland?