Disclaimer: I am not a historian, however I do have a passion for history, and have researched some of the theories regarding the origins of my language and culture. If my answer doesn't meet H SE standards please let me know and I'll try and improve it (it's my first post)
I noticed this question in the HNQ list and joined the site simply so I could get my two cents in. As my name might clue you in, I am Romanian.
The question you ask is one that is actually being debated more and more in recent years. The traditional explanation is the one which Semaphore offers above: Romania was a Roman province, and we were simply influenced by them.
This explanation is the one that I was taught in school, and the one that my friend, who holds a Doctorate in History, was also taught throughout his academic career. However there are many historians coming forward now-a-days who question that explanation. Unfortunately, these people seem to be regarded as rocking the proverbial boat (as per my friend, who works for the Romanian National Historic Institute), and are mostly discredited within Romanian circles, although less so abroad.
There are two aspects to them doubting the official story:
Adoption of Language?
Modern day Israel was a Roman province for well over 400 years. Over that time the Romans tried to impose their rule over the locals quite ruthlessly, yet their language was not at all affected by those centuries of Roman rule - at least, not fundamentally.
Dacia was a Roman province for less than 200 years, however the Romanian language is so close to Latin that as a 13 year old schoolboy I could read Latin texts and understand the gist of them with no language training whatsoever. Our verb "to be" is conjugated the same as in Latin. The similarities are nearly endless.
Furthermore, Romanians are separated into 3 major regions: the Romanian Plains, or "The Romanian Country", as we called it, which is the region south of the Carpathian mountains, Transylvania, which is the region contained within the Carpathians, and Moldova, the region to the right of the mountains, and which is now split between Romanian territory, and the independent country of Moldova (thanks to the Soviets).
Over that vast area the Romanian language, although peppered with small differences in vocabulary and accent, is fundamentally identical. However, the Romans only ever conquered a relatively small part of Romania, in the south-east. Most Dacian tribes were not, in fact, under Roman rule, and, especially in those days, would not have come into direct and constant contact with the Romans (people would be born, and die without having ever left an area of a few square kilometers).
The official theory is that the Romans colonists simply influenced the local language. However, how could that have happened so completely, over so wide and area, and influenced all Dacians?
Furthermore, most Roman colonists were legionaries, and while Roman in theory, in practice they came from many different backgrounds, spoke many different languages, and communicated with each other mostly in Pig Latin. This raises yet another question: why is Romanian so similar to classic Latin?
The counter argument raised is that Roman merchants would have traveled to those communities and thus introduced the language to them, however, realistically, this same phenomenon completely failed to influence the language and culture of other people's, such as the Germanic tribes.
Links to the Past
Last but not least, even today, us Romanians use some expressions the origins of which can be traced back many thousands of years, to the worship of the goddess Gaia (which the Dacians worshiped), and other Dacian practices.
"Să te ia Gaia", or "Lua-te-ar Gaia" is an expression which my grandparents can remember their own great-grand-parents using. It literally means "May Gaia take you". The origins of this expression are so ancient as to send shivers down my spine, yet it is still widely used today. Had Roman culture and language overtaken Dacian language and culture so completely surely this expression would not have survived such a change?
"L-ai făcut pe Dracu ghem" - this expression, which exists in a number of different variations, translates to "You folded the Devil into a ball" (used to mean "you screwed up"), which is descriptive of the Dacian warrior's habit of rolling up their battle standard - a dragon with the head of a wolf - and running the heck away when defeated on the battlefield. Again, this expression links back to a behavior so ancient as to be breathtaking. Something which has not been done for well over a thousand years, yet is still part of our cultural memory today.
This might not seem like a big deal to some people, but to me it's indicative of my language's deep, deep ties to the past - and they are not the only such "hints". Had we been so thoroughly Romanized that we readily abandoned our language and culture we would not, IMO, and in the opinion of these historians, be saying these things today.
Other Historical References
As I mentioned, the Romanian people are geographically divided into three distinct areas. These areas were - throughout our history - under the influence of one major empire or another.
We were - historically - trapped between the Turks, the Autro-Hungarians, and the Russians, not to mention the Tatars, who also made their presence felt in Moldova.
These empires were much more powerful than the three Romanian countries, and used us a buffer. Many bloody wars between these powers were fought on Romanian soil, and each Empire made sure - in their own ways - that the Romanian people did not unite and become powerful ourselves.
In 1600 Michael the Brave made a very well known attempt to unite the three Romanian countries. The Turks had gained significant influence in the Romanian countries, and this worried the neighboring emperors. Michael received backing from the Austro-Hungarians to drive the Turks from Romanian territory, which he did. He fought them across all three territories, and was received as a hero by all three Romanian countries. He was crowned their king, at which point his allies betrayed him. The Austro-Hungarians sent executioners to meet with him, rather than a diplomatic party. He was summarily executed by decapitation in front of his own command tent.
This historic event is well remembered in our history, but less so are some of the letters between Michael the Brave and other Romanian nobles of the time, in which he spoke not of creating a unified Romanian country, but of uniting Dacia.
And he is not, by a long shot, the only one to reference Dacia, and the Dacian people, rather than Romania, or the Romanian people. Other very famous historical figures from our history also speak extensively of Dacia, and our heritage as Dacians, not Romans, or Romanians.
Not long after this attempt at unification, worried by the influence that the Turks had achieved, the Catholic Church initiated an exchange of clergy between Transylvania (under the control of the Catholic Austro-Hungarians), and Rome. These Orthodox priests and clergymen were invited to study in Rome under the pretense that Christians should stick together against the Turkish influence.
It is at this point in time that Romanians start referring to themselves as such, and that notable historical references to us being the descendants of Romans begin to enter our official documents.
It is the belief of these historians that a monumental propaganda effort was made to insert this belief within the history of the Romanian people in order to strengthen our bonds with Rome, and thus the Christian fate, as we were in direct contact with the Turks and had to be counted on to keep them at bay.
Such papers which I have read - which are automatically dismissed in Romanian historical circles - show, in my opinion, pretty good evidence of this trend in our history.
And so, if we are not the descendants of Romans why is Romanian so close to Latin? This is the newly proposed theory (in a few short sentences):
The Dacians were, at their origin, Thracians. So were the Trojans, and the survivors of Troy are known to be the founders of Rome.
What these historians claim is that rather than Romanian being a child of the Latin language, they were cousins long before the Romans invaded Dacia. That the Dacian and Roman people were related from ancient times, and that this is the reason why our languages are so similar - both are of Thracian origin.
I'm going to end my answer here, as it has already grown to a rather ridiculous length. I know that I'm not providing sources for the above, however everything is easily Googled. I can provide some references later today (at work at the moment), if requested.
I personally, as a child, was a great believer, and very proud, that we are the "descendants of Romans". In recent years, having read of this theory, and looked at some of the facts which these "rogue" historians are pointing to have grown to be very doubtful of the official story.
In the end, this all happened so long ago that we may never know for certain.