Unsurprisingly, this question is on the mind of basically everyone who studies late antiquity. A funny way of looking at the question is this list, which lists many of the reasons people have suggested to have caused the 'fall' of 'Rome'. A lot of them are quite bonkers, but together it does give you an idea of how complex this whole process was. For myself, I have to begin by noting that Rome did not fall. 476 saw the deposition of a rather powerless puppet emperor, Romulus Augustulus, by one of his generals, Odoacer. This general then promptly submitted (on paper at least) to a previously overthrown emperor then ruling in Dalmatia; when this man, Julius Nepos, died, Odoacer threw his lot in with emperor Zeno, the Eastern Roman emperor ruling in Constantinople. Again Odoacer wasn't really loyal to Constantinopolitan interests, but what this move represented is clear enough - the Roman Empire did not end, but continued to exist for many people. 476 was certainly an important year, but it did not by any stretch of the imagination represent the collapse of civilisation. It was only in the 520s that we have a source proclaiming 476 to be the year the Western Roman Empire had ended, and the chronicle, that of Marcellinus Comes, was composed at a time when the emperor in Constantinople was willing to contemplate an invasion of Italy again. In some sense, 476 can be seen as a 'manufactured turning-point' that was created to suit the political circumstances of the sixth century rather than a date reflecting the reality of the fifth century, one that obscured just how much continuity there was between the empire of 476 and the kingdom of Odoacer as well as the pseudo-imperial Ostrogothic Kingdom of Theodoric the Great, the 'barbarian' ex-consul of Constantinople who had overthrown Odoacer in 489.

Even disregarding this, the Roman Empire lived on the east for nearly another thousand years, enduring crises that far outweighed the problems the empire faced in the fifth century. An epoch-defining war with Persia lasted for 25 years in the early seventh century, a calamity that was swiftly followed by the Arab conquests, which ushered in a new age of 'Mediterranean total war' and the destruction of the old order. For centuries afterwards, the Romans had to deal with persistent border wars with its Arab and later Turkish neighbours, all the while wracked by civil wars and devastating religious controversies. Throughout all of this, the Romans maintained their culture and faith, a fact recognised by pretty much all their neighbours. Even in the post-Roman West, Romanness was valued by the 'barbarian' kings who carved out new states in its ashes. Kings such as Clovis of the Franks and Gundobad of the Burgundians hungered for imperial titles granted by Constantinople, whilst other rulers such as Reccared of the Visigoths and Aethelbert of Kent were lauded as 'New Constantines' by their admirers. The same was true for the Arabs who after the 640s ruled the Roman Near East. In so many ways, the Romans did not fall, but endured both as a political entity in the East and in terms of its cultural legacy as a major phenomenon for centuries after 476 throughout the post-Roman world.

But to come back to the main question, there are a few important factors worth sketching out. I think one of the most important factors to remember here is that the men who did the most damage to the stability of the Western Roman state, such as Alaric of the Goths who sacked Rome in 410 and Genseric of the Vandals who sacked Rome in 455, were not 'barbarians' unfamiliar with the empire, but instead men who had lived within the empire for decades and knew perfectly well what the empire's strengths/weaknesses were. Moreover, they were able to participate fully in the brutal world of Roman politics, creating and bringing down emperors at will, just as Roman generals were wont to do throughout the empire's history. These 'barbarians' were new actors in an age-old story, men who wanted to seize political power to achieve their own goals. 'Real' Romans sometimes opposed them, but more often they tolerated and actively embraced them in their own machinations. Ambition was not a unique trait for any group of people. As a result of these power-plays, the frontiers were weakened and the central government made less influential, as seen in the withdrawal of troops from Britain, the making of expedient deals with 'barbarians' to get manpower quickly and the rise of regional elites willing and capable of carving out a piece of the empire for themselves. These regional Romans and their 'barbarian' neighbours were no less sophisticated or talented than the 'real' Romans, so it should not be a surprise that they were able to contest the authority of the emperor.

These problems were compounded by two other factors. The first was surely the rise of the Hunnic Empire in the fourth/fifth centuries, which first pushed many peoples, such as the Goths, into the Roman Empire and later directly confronted imperial forces. The impact of the rise of a new superpower in the region cannot be underestimated. Just as the Roman centre faced internal unrest, it now had to confront a new unprecedented military threat. The Eastern Roman Empire was always wealthier and was able to buy the Huns off, though even then they faced problems, but the Western Roman Empire wasn't so lucky. This was partly due to the other major crisis facing the empire, one of a financial nature. The North African provinces were fabulously wealthy, but they were lost to the Vandals in the 430s and from then on the empire struggled to recover, as its one remaining relatively stable taxation base was lost. The empire was now no longer able to pay for significant armies under its direct control to operate against its foes, as instead it often delegated its responsibilities to regional 'barbarian' kings, who sometimes co-operated with Roman interests, sometimes not.

Even so however the Western Roman state managed to limp on for many decades, but the final nail in its coffin was in 468, as a combined Eastern and Western Roman expedition against North Africa was decisively defeated in that year. Without the resources of North Africa, the western empire could not sustain itself and its last decade saw numerous revolts by soldiers complaining about their pay, one that finally culminated in Odoacer's revolt in 476. For further reading, consider Adrian Goldsworthy's The Fall of the West (2009), Peter Heather's The Fall of the Roman Empire (2005) or Guy Halsall's Barbarian Migrations and the Roman West, 376–568 (2007). For a book emphasising the economic decline of this period, check out Bryan Ward-Perkins' The Fall of Rome and the End of Civilization (2006). For an understanding of how much continuity this post-Roman world still had with the empire before 476, Chris Wickham's The Inheritance of Rome: A History of Europe from 400 to 1000 (2010) and Peter Sarris' Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 are both great reads.

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