What happened to the Goths? In a nutshell, they were absorbed into other ethnic groups as the reason for affiliation with Gothic-ness disappeared with the decline of their power. In a lot of ways, this is exactly the reason for the disappearance of Romans, as that identity was co-opted by desires to enter into the ethnic groups of their successor nations.

You have to keep in mind that the way most people think about ethnicity is a peculiar 19th century construct that attempted to project an eternal essentialism into the far past. It presumed that people spoke the same unified language, had the same unified culture, and didn't deviate from born-in cultural behaviors. Modern research into identity has proven this to not be the case, and you can see the test cases all around you and in yourself considering what identities you "choose" at particular moments (race, age, sex, class, politics, hobbies, etc). Though there are impediments to the crossing of identity boundaries, especially in the cases of elite identities, they are over time fluid, situational, and adaptive.

As an example, you should consider that from a "genetic" standpoint, it's very plausible that Italians descended from families that only stayed put in what is now Tuscany in Italy could over the course of 2500 years of Italian history consider themselves Etruscans, Romans, Goths/Lombards, Franks, Tuscans, and Italians. This by the way has actually been demonstrated with genetic testing done on the ancestry of modern Brits, in that they're still just as "Roman/Celtic" in ancestry as ever, despite germanic/french invasions. Only the identity and cultural label changes. Nothing about "blood" changes.

Now as for why desire to be Gothic would cease, that's pretty much straightforward military defeat. The Ostrogoths as an elite hierarchy disappeared after their loss to the eastern Romans in the Gothic war and subsequent Lombard invasion replaced them twice over as the ruling group. The Visigoths as an elite hierarchy disappeared after their conquest by the Arabs.

Why did the Franks rise to become an ingredient in modern France? Because they stuck around. They were never conquered/replaced as an elite identity (for a long enough time anyway) in France.

tl;dr - People want to become the identity of "ethnicities" possessing power. As those ethnicities cease having power, those ethnicities cease being attractive. Over a long enough time, those losing ethnicities disappear. Well, until 19th century nationalism comes along, but that's a whooooole other story.

Sources:

• Geary, Patrick J. The Myth of Nations: The Medieval Origins of Europe. Princeton, NJ: Princeton UP, 2002. Print.

• Innes, Matthew. Introduction to Early Medieval Western Europe, 300-900: The Sword, the Plough and the Book. London: Routledge, 2007. Print.

• Wickham, Chris. Early Medieval Italy: Central Power and Local Society, 400-1000. Totowa, NJ: Barnes & Noble, 1981. Print.

Part 1: This is an unpleasant way to answer, but I'm commenting here so I can get back to you later.

Part 2: Hi there! I'm back; and I'd like to preface by saying that my area of expertise is North Italy in the communal era, but I think that late antiquity and the very early middle ages are a fascinating time period; much more interesting than the "Castles and Knights" mish-mash of late-medieval and early renaissance preconceptions that reigns in western media. I'll answer you as best I can, but I'm sure that an expert in the migration period might know more.

The influx of barbarian migrants into the Western Roman Empire was less the barbaric horde most people believe it to be, and more of a gradual, systematic integration of Germanic tribes into Roman society; especially into the army, the longstanding gateway for Romanization.

In all this, the Goths were a pretty strange tribe. After having wreaked general havoc in the Bankans and some parts of Anatolia fairly early (in the 3d century!) they first settled in modern day-Romania, but were eventually pushed south during the Hunnic Invasion; being absorbed into the Eastern Roman empire as Foderati, or a tribe that had the privilege of living within Rome's borders in exchange for regular military service.

The union wasn't "Clean" though; a historian of the Migration Period will be able to go into detail about the split between Visigoths, who moved south and became Foderati, and Ostrogoths, who submitted to the Huns and lived as their vassals. However, the Ostrogoths eventually revolted agains the Huns and also "joined" the Eastern Roman Empire (effectively settling on the opposite side of the River Danube, charged with defending the river against the Huns).

The Visigoths, who had joined the Roman Army, participated in the general murdering, backstabbing, and plotting that characterized the late Roman Empire, as different factions within the army fought, plotted, and caused general havoc as the empire collapsed around them. Through this general chaos, the Visigoths carved a kingdom for themselves in Iberia in by 5th century, stretching from modern Portugal as far as southern France. This is another part of history I would like to be more familiar with; I have read very little of this first Iberian Kingdom, but as it was eventually conquered by the Umayyads, the final days read rather like a fascinating apocalyptic log. After the death of the last Visigoth King in battle in 714, a nobleman called Pegalus rounded up what resources he could and founded the Kingdom of Asturias, which would slowly reconquer Iberia from the "Moors" and develop into modern Spain and Portugal. You really couldn't make this story up if you tried.

But what happened to the Ostrogoths, who we left on the banks of the Danube? This is where the story gets interesting, answers your question, and I can get a lot of detail.

Having been not only "Absorbed" into the Eastern Roman army, but also making up a significant portion of it, the Ostrogoth generals were a powerful party in the Byzantine court (when they weren't arguing with each other). When Zeno came to the throne in 474, his only real claim was that he had married the previous emperor's daughter. The fact that he was from modern Turkey also didn't help his case either. However, with the help of a popular young Ostrogoth general named Theodoric (who's father had led the Ostrogoths in revolt against the Huns), Zeno was able to crush his enemies and win support of the army (in the process, defeating another Ostrogoth general, also named Theodoric).

Zeno, however, saw that Theodoric was in a position to either strongly influence his policy, or outright overthrow him (he did command the loyalty of most of the army, after all). So he hatched a plan to send Theodoric on a mission: Conquer Italy.

As successive parts of the Empire slipped out of Rome's grasp, all that was left was Italy. By this time, the Emperor in Rome amounted to whomever was able to convince the most of army to back their claim (and defeat the part of the army that didn't). Indeed, the fabled Odoacer didn't really end the Roman Empire: he was just one in a long line of thugs who grabbed the throne, he was just the first to give up the schtick of pretending to be Roman Emperor. Indeed, countless emperors before Odoacer had been of "Barbaric" descent, Odoacer himself was fairly romanized (his first name was Flavius; not a very run-of-the mill barbarian name!).

By sending Theodoric and the soldiers loyal to him in Italy, Zeno achieved two things: First, getting rid Odoacer, who could become a potential enemy, even if he had acted pretty docile up to then (he even sent the imperial paraphernalia to Constantinople, however, he had the Roman Senate firmly in his grasp) and second, getting rid of pesky Theodoric.

Setting out in 488, Theodoric seized northern Italy fairly easily, defeating Odoacer at the river Isonzo, and later Verona, but soon both sides became entrenched on opposite sides of the river Po, Theodoric at his stronghold in Pavia, and Odoacer at Ravenna; a difficult city to siege as it lay in the middle of a swamp. Odoacer broke through in an offensive in 490, taking back Verona and Mediolanum, but just as Theodoric was hemmed in at Pavia, the Visigoth King, Alaric (who happened to be married to Theodoric's sister) showed up at the eleventh hour and beat Odoacer back. Again, you couldn't make this up if you tried (this is why I love history. Who needs fiction?).

Theodoric and Alaric tried laying siege to Odoacer in Ravenna, but the city is easy to supply from the sea, and the Goths had no navy. The siege dragged on and on, until the Bishop of Ravenna brought the two parties together to try to negotiate a truce. Theodoric and Odoacer eventually agreed to make peace: they would rule Italy together, both as Consuls. After all, Odoacer had never declared himself King of Italy, he had only forced the Roman Senate to declare him Consul, and as per Roman Republican tradition, there were always two consuls.

Then, that night, at a banquet celebrating the peace, Theodoric murdered Odoacer.

Emperor Zeno promptly declared Theodoric King of Italy; on paper, he was a vassal of the Eastern Empire, but de facto, he was independent. Zeno got what he wanted; a useful buffer state to the west, a troublesome individual out of his country, and a friendly Italian kingdom.

The Ostrogothic Kingdom, with its capital at Ravenna, would continue to exist for some sixty years, maintaining many of the same traditions of late Roman society. Theodoric himself had grown up at the court in Constantinople, Greek was his first language, and he was fluent in Latin (reports on his ability to speak Gothic vary). Most important goths emulated Roman lifestyles, and public administration was mostly still run by native latin speakers.

Then, in the mid-6th century, the Lombards invaded Italy, but that is a story for another time.

If I've piqued your interest in Theodoric's fascinating life (and I hope I have!) you should check out a few books on him! I most certainly couldn't include everything! My favorite english source (which I have recommended multiple times on this sub) is Theoderic in Italy by J. Moorhead (University Press 1992). Sorry for the long post. I just think that the early middle ages are a very underrated time period.

I know that the Visigoths had a kingdom in Southern France. They were pushed south by the Franks and set up a kingdom in Iberia. When the Muslims invaded, the Visigothic kingdom crumbled and some Visigothic nobles retreated to the Asturian mountain range in the north creating the Kingdom of Asturias which would eventually become the kingdom of Leon, which was then incorporated in the the Kingdom of Castille. I believe the Visigoths eventually came to identify them as Castillian over time as identities changed. I can't speak as much for the Ostrogoths in Italy though.

Edit: Added a sentence