There are several answers to that question; various authors have favoured one or another, but it is probable that the fall of the Western Roman Empire was due to their combination.
From a geostrategic point of view, the stability of the Roman Empire was guaranteed by the legions: strong but not numerous forces, able to intervene in many places thanks to their high mobility. Legions were thus supported by strong logistics (supply chain) and infrastructure (the famous Roman road network). However, the West, being less populated than the East, had comparatively more boundaries to handle with less troops. Moreover, a big part of the Eastern Empire boundaries were shielded against invaders by geography (Black Sea, Caucasus mountains, Arabic desert...) and by the Persian Empire.
In the fourth century AD, due to increasing population, the "Barbarians" were seeking land to settle. From the point of view of a Visigoth, Rome is the Civilization, and room was to be found mostly in the Western part of the Empire. Indeed, all along the third and fourth century, there had been a constant stream of "germanic" newcomers who were willing to become part of Rome, and eager to obtain land, by charity or by force, whichever was necessary. What changed at Adrianople is that the immigrants became too numerous to content with henceforth unoccupied land, and they turned out to be too strong to be dealt with militarily. Adrianople is the symbolic pivot because in the aftermath of the battle, Rome had to evict Roman citizens to meet the demands for land of the invaders. This is the point where Rome failed to defend its own. This is the moment where people began to cease to believe in Rome.
Demographically speaking, the fall of Rome, or at least part of the Empire, was "unavoidable" (with all the usual caveats in these matters), since the population in the Empire was somewhat constant, while it was rapidly increasing in northern and eastern Europe. That the West half should fall first was "logical" because it was larger, less densely populated, less supported by its economy and infrastructure, and primarily targeted by invaders.
(Edward Luttwak's books are a very good reading on the strategic aspects of the Roman Empires, both before and after the split: The Grand Strategy of the Roman Empire and The Grand Strategy of the Byzantine Empire.)
These considerations must not hide other facts:
From the third century onwards, a constant theme in the Roman Empire was regular usurpations. Most emperors got the job by being proclaimed by their enthusiastic legions; when such a general was given this "nice" present by his troops, he had little choice but to roll with it and try to defeat the other legions still loyal to the previous emperor. If he succeeded, then he knew that he would have to crush many similarly rebellious commanders. The Roman military forces were mostly employed to quell such unrest.
In the late fourth century AD, it had become quite impossible for distant provinces to obtain any form of military help from Rome. This is the meaning of Rome's evacuation of Great Britain in 410 AD: as Honorius puts it, Britain's cities had to fend for themselves now. This was what they had already done for some time: legions were sent only to suppress rebellions. In that sense, the Western Empire can be said to have fallen by fighting itself to death.
Germanic people were increasingly part of the legions themselves. Rome was finding expedient to hire these newcomers, who were more easily expendable than Roman citizens. Great Germanic generals like Stilicho and Odoacer were really considering themselves as "Romans". From their point of view, they did not "invade" the Empire; they were the Empire.
It has been argued that a severe and long-standing economic crisis had made the Empire unsustainable. The roots of that crisis have been variously attributed to demographic stagnation, rarefaction of precious metals, disruption of long-distance commerce... even epidemics, introduction of rats and climatic changes have been invoked. In fact we don't really know; but we can see, for instance, that the population of Rome (the city) had been steadily decreasing, from a maximum of about 1.65 millions in around 100 AD, down to 1.1 millions in 400 AD, then sharply falling to 0.5 millions in 450 AD, then 100000 in 500 AD. Though the big drop is contemporaneous with the formal "fall of Western Empire" of 476 AD, the problem was apparently much older.
Therefore, it may be so that Odoacer, once having grabbed the reins of power, was lucid enough to realize that the former model had outlived its usefulness, and had to be dropped. Such economy-related explanations may make sense only if we can explain why the East was able to pull through; since climate, agriculture and commercial networks in the East used different structures, this kind of explanation is possible.
Some authors have blamed Christianism, preventing proper assimilation of Gothic people because they are Arianists, due to an historical accident: Ulfilas converted them to Christianism just at a time when Arianism was still fashionable. Goths and Lombards have clung to their now heretic liturgy because it was in their own language. The intransigence of Catholic rulers would have triggered rebellions, then chaos and fall.
This is too simplistic a story to be accurate. However, it highlights the fact that the society structure was changing. Indeed, the Germanic people were trying to become Roman, but not the same kind as the Romans already there; they still wanted to retain some of their identity. In very anachronistic terms, we might say that the new immigrants were not content with a central government, and were pushing for diluted federalism.