Why did the Romans change Europe's language, but the barbarians didn't?

The main difference is that the Romans settled and "colonized" conquered territories, and the barbarians often did not.

For instance, after Caesar's victory at Alesia against Vercingetorix, the Romans sent administrators and garrison troops to integrate Gaul into the Roman empire, even though Caesar himself was otherwise occupied playing Roman politics. The same was true with Spain after Scipio Africanus' victories two centuries earlier.

Then there was the Roman policy of "all roads lead to Rome." Although this technically referred to geography, the Romans followed similar policies with language and culture. Leading men of conquered territories were encouraged to become Roman citizens, further contributing to the diffusion of Roman language and culture. Basically, Latin was the means by which members of one part of the empire communicated with members of other parts of the empire speaking a different "native" language, for trade or other purposes.

On the other hand, the barbarians made few such contributions. Many of them were out for plunder, and backed away after they got what they wanted from western Europe. Attilla the Hun was one such example. Others came to stay, but were assimilated by the local (read, Roman or quasi Roman) language and culture, e.g. the "Franks".

As a complement to the most voted answer that already says that the problem is complicated. I will try to give a "simplificatory" answer.

The question is based on a misleading image of Latin dominating other languages with which it competed.

A few facts need to be pointed out:

Like Summerian, Assirian, Ancient Egyptian, Ancient Chinese, and Classical Greek, Latin is not simply the language of some people that happened to conquer an empire, but it is the main tool or part of that very centralized state, and of its administration.

Outside the Greek speaking world, in pre-Roman Europe there was never a long-lasting well-established statehood which would have triggered the need for a central administration ruled by a common language (with maybe one exception, the Etruscans.)

Latin was a language of political intrigue, army and (high, literate) culture. As such, Latin was not in competition with local, tribal, vernacular languages, which sometimes survived all over the former territory of the empire until the 20th century, in most cases (with the exception of Celtic, Basque and Albanian) mutating in local vernacular forms of Romance idioms that are the real ancestors of the present Romance languages. Those are not the direct descendants of the language of Cicero, but of local popular and different forms of Latin.

As a language of administration etc Latin was not rivaled by the languages that it has "replaced": that kind of administration, including language-administration, was simply absent there before the Roman conquest. In a sense there was nothing to replace, and what was there was not really replaced, but only transformed. In areas where other languages were already operating at a higher level, like Greek and Aramaic, these were NOT replaced, or they were replaced only partially. (The only cases that come to mind of a high-status language replaced by Latin are Etruscan and Carthaginian.)

The vernacular languages were not in fact languages in the sense Latin was. They were not written, not standardized, not supported by a strong self-perpetuating administration. Ideas like that of a Celtic language common to all Celts, or just to Gauls, is just a vague hypothesis with no real factual base. The same is true for various other cases, like the Macedonian, Thracian, Dacian, Illyrian. Only in such areas where we have little to no knowledge of the pre-Roman language are we tempted to hypothesize a local standardized language on relative large territories: where we really have some knowledge of ancient linguistic areas, like in Italy and Germany, we see an enormous diversity. Latin was the language of a very small part of Italy. It took the whole might of the Roman arms to impose Latin as the Language of Italy. The fact that to this day there are a lot of Italian vernacular idioms (and there were many more 100 years ago!) shows the limit to that standardization even there. (The same is true for France and Spain.)

Nobody imagines a common language to Italy before the Roman conquest. But (more or less nationalist) historians argue about one common language of the Dacians, one of the Thracians, of Moesians, of Illyrians, and about whether they were the same or related, or not. Probably none of those peoples, albeit related, had ONE common language, until the Romans came.

Roman conquest provided Latin as a common language in a sea of local idioms, just like Aramaic was the language of the Achaemenid Empire (and Koine Greek after Alexander of Macedon). Latin has not resulted in local vernacular Romance languages in areas where the need for a such lingua franca was already provided for. (A such scenario may bring light on obscure problems of language history like that of how was Dacia romanized in just 170 years. French and British colonization of Africa was shorter but people there have adopted those languages for the simple reason that these were useful as a lingua franca. Swahili plays a similar role in parts of Africa. And another similar example is Arabic.)

As for why the Germanic and other invaders did not replace Roman idioms, the answer is that they DID as much as they could. That is, especially, when they had one unified, known by all language to impose. Many invaders were unions of different tribes of various origins and sometimes they spoke some type of vulgar Latin in order to understand each other and especially to be able to be understood by the people they had just conquered. Language was a practical matter for them, not a matter of prestige. And, as far as prestige is concerned, Latin had no rival excepting Greek. Maybe sheer numbers could have tipped the balance the other way: but in most cases the invaders were in minority and adopted the language of the conquered peoples. — A case where "Barbarians" had the numbers on their side would be the Slavs. They have flooded the eastern part of the empire (beside the rest of Eastern Europe) and became the majority in areas that they didn't even controlled politically. When the Turkic Bulgars conquered them and created a big empire in the Balkans, they adopted Slavic in a relatively short time, which means that Slavic-speakers might have dominated numerically future Bulgaria. (In fact things were even more complex: Wikipedia states that "The South Slavic group, despite sharing a common language, is separated and has a largely different genetic past from their northern linguistic relatives. Therefore, for the Bulgarians and most other South Slavs the most plausible explanation would be that their most sizable genetic components were inherited from indigenous Balkan pre-Slavic and pre-Bulgar population." In that scenario South-Slavic became the language of the first Bulgar empire because it offered a common language to its new and older populations that before spoke different languages: Latin, Greek, Thracian, Turcik. It is also interesting that "Despite various invasions of Altaic-speaking peoples in Europe, no significant impact from such Asian descent is recorded throughout southern and central Europe".)

There were big differences between various areas of the former Roman empire: some areas were populated by peoples that already spoke a Romance idiom, some were not; some of the previous residents were more numerous than the invaders, some were not; some territories lost any previous centralized administration, others did not, or did to a lesser extent. The Latin-speaking Catholic Church administration, and the Greek-speaking Byzantine administration were big players in perpetuating central administration, but population numbers must have played the biggest role as far as local, vernacular languages are concerned. (And some of these idioms, depending on historical accidents that placed the main centers of power and culture in one country's linguistic area rather than in another, are the base for modern Romance languages — Parisian French for France, Tuscan-Florentine for Italy, Castilian for Spain —, not classical Latin.) Depending on these factors Germanic languages conquered the British Isles, Slavic languages spread in the Balkans, Hungarians imposed Hungarian to Pannonia, but Italy, Iberia, Gauls and (as surely but more mysteriously) Dacia ended up with idioms of Latin ancestry, which, on the other hand, are all different, and heavily influenced by the languages of the "Barbarians" (of those that preceded the Roman conquest as well as of those that followed it).

This answer is inspired by an idea (applied to Dacians) presented in a book by Dan Alexe — a book that I guess is only available in Romanian: "Dacopatia şi alte rătăciri româneşti" (Dacopathy and other Romanian delusions).

Another factor:

When Rome moved in on the Eastern Mediterranean it found a lot of pre-existing civilisation: Hellenistic Greek-speakers, Aramaic speakers, etc.. The Romans worked through these existing civilised elites because it was less work than setting up everything themselves. As a result, these places mostly retained the languages of the pre-Roman elites. (At least until Islam arrived, with religious reasons for making everyone learn Arabic.)

But in Western Europe there wasn't a lot of local civilisation to work through. Tunisia, Libya and parts of Spain were civilised by the Carthaginians but Rome got along so badly with Carthage that it ended up trashing all that. Maybe that's what happened to Etruscan as well (the rest of Italy south of the Po was mostly speaking languages related to Latin anyway). There were some Greek colonies and parts of Gaul were a high-end barbarian culture knocking on the door of civilisation. But mostly the Romans had to build civilisation from scratch, so they set up Roman elites who created Latin-speaking civilisations.

When the Germans moved in on Western Europe they found Roman elites and worked through them. And so Latin survived long enough to diversify. An exception was Britain, where the war with the Romano-British went on long enough that the Romano-British decivilised from the stress and when the Anglo-Saxons got interested in civilisation a few centuries later they had to build a new civilisation with only limited assistance from the British.