The population of Rome (the city) and the whole Italian peninsula did not suddenly collapse during the “barbarian invasion” associated with the fall of the Western Roman empire in 476 or even in it’s immediate aftermath. If we focus at the population of Rome specifically (only city in the Italy peninsula with a population in the 300 - 400 thousand range) it’s has been steadily decreasing since the 2nd century. It’s estimated that at it’s peak in the 2nd century (when more than 1 millions people might had been living in the city) 25% of all it's population was directly relying on the free food provided by the government, 25% were slaves and another 10% were employed by the municipal or imperial government apparatus. The high population could only be sustained through huge economic transfer from other provinces.

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So as Rome lost it’s political importance when the center of political power shifted to northern Italy and Constantinople and economic ties between provinces began weakening the population naturally decreased as it’s inhabitants were forced to more rural areas of Italy and to other provinces.One thing that’s important to understand is that the “barbarian invasion” of Rome was not some sudden out of the blue event akin to the Mongolian invasions of Eastern Europe (where large hordes of Barbarians appeared seemingly out of nowhere and proceed to conquer and sack all the cities in their path). Romans had been interacting with most of the groups that came to eventually rule various parts of the Western Empire like the Goths and (to a lesser degree) the Franks, Saxons and Vandals for several hundreds of years. By the start of the 5th century around a quarter of all officers in the Roman army were born as barbarians. And due various issues related to recruitment and financing of its armies the empire increasingly and eventually almost entirely relied on foederati units provided by “vassal” barbarians tribes in exchange for the right to settle in Roman territory. Often these barbarians rose to very high positions in the military hierarchy, like Stilicho (a Vandal general who basically a de factor ruler of the Western Empire between ~400 an 408). So when Flavius Odoacer (an officer in the Roman army and the commander of all foederati troops in Italy deposed the last Western Roman Emperor in 476 and declared himself the King of Italy it wasn’t a huge shock to most Italians. The Goths established themselves as a military ruling class but mostly allowed the native Romans to govern themselves, the Gothic king still funded traditional Roman festivals, Roman law was still maintained and there is some evidence that the authority of the Roman senate increased considerably under Gothic rule.

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The sudden demographic collapse which left Rome itself basically a ghost town and resulted in significant depopulation across the entire Italian peninsula was actually a result of a result of an attempt by the Eastern Roman Empire to reconquer Italy from the Goths. During the Gothic Wars 535 - 554 following multiple sieges in 537, 546, 549 as the city changed hands between the Romans and the Goths its population collapsed to ~ 30,000 people. Most large cities in Italy declined significantly during the war since it was no longer possible to supply then due to constant warfare and devastation in the countryside, Milan (which was capital of the Western empire between 286 and 402 and had a population of 100,000 and at the outbreak of the war was the second largest city in Italy) was almost completely destroyed (according to ancient source all male inhabitants were killed and the city was razed to the ground). This was also significantly exacerbated by the Plague of Justinian which began in 541 and killed from a quarter to a half (depending of estimated) of all people in the Mediterranean region.