What happened in Rome, when the western empire "fell"?

By 476 the city of Rome had not been the center of the Empire for a long while. Diocletian, picking up the pieces after the crisis of the third century, made a point of snubbing Rome compared to the "actual" imperial capitals, which he considered more important. Moreover the sackings Rome experienced in the fifth century certainly depleted its riches and resources. From the Cambridge Ancient History, vol. 14, page 378:

Despite considerable imperial and Ostrogothic attempts to sustain it, Rome gradually shrank through the fifth and sixth century to a tiny proportion of its former size; not only was there no longer the government money to sustain it, but it also inevitably lost out from the shrinking overseas holdings of its senatorial aristocracy.

Doubtless the Roman Senate still sitting in Rome (now no more than an anachronistic relic) was the theater of some speeches, but the historians of the time deemed them of so little consequence that they don't even mention them. But even beyond Rome, in the imperial capital of Ravenna, and all the other lands nominally under the suzerainty of the west Roman empire, the news would have been of little consequence. This passage of the Cambridge Ancient History (vol. 14 page 25) is illuminating:

Since the empire’s financial base was simultaneously decreasing, the idea of empire quickly became meaningless. The centre no longer controlled anything anyone wanted. In consequence, the late 460s and 470s saw one group after another coming to the realization that the western empire was no longer a prize worth fighting for. It must have been an extraordinary moment as the realization dawned on the leaders of individual interest groups, and upon members of local Roman landowning élites, that, after hundreds of years of existence, the Roman state in western Europe was now an anachronism.

and ibid., page 27

A subordinate commander, Odoacer, organized a putsch, murdered Orestes, and deposed Romulus, derisively titled Augustulus, on or around 4 September 476. He then sent an embassy to Constantinople which did no more than state the obvious. There was no longer any need for an emperor in the west.

Also, from Christopher Wickam The Inheritance of Rome (a book on Early Medieval Europe that I heartily recommend to those interested in that neglected period)

Italy is actually the region of the western empire which lived through least change in the 470s, for Odovacer ruled much as Ricimer had, at the head of a regular army. Italy did not experience an invasion and conquest until 489–93, with the arrival of Theoderic the Amal and his Ostrogoths, and Theoderic (489–526) ruled in as Roman a way as possible, too.

In addition to what I wrote, the "fall of the Western Empire" is a bit of a misnomer. It was not, at the time, perceived as a change in political order: all the lands formerly in the Roman Empire were still under the nominal authority of the Emperor in Constantinople. The West versus East division was, at least on paper, only an administrative one, and the two Emperors had theoretically been coequal heads of state, with the same authority on every component piece. The situation on the ground was very much different, but Odoacer mantained the polite fiction of being the "viceroy" of the Emperor in Italy and even recognized and minted coins in the name of Julius Nepos, the nominal new Western emperor. The loss of authority of the Roman Empire over what were formerly its Western lands was a slow and gradual process, and while 476 is a convenient date (the last year in which a sitting emperor resided in the Italian peninsula), it's quite arbitrary.

Other dates that could be reasonably advanced for the loss of imperial control of Italy are 572, when most of Italy was conquered by the Lombards, a power that did not recognize even the nominal suzerainty of the Emperor, or 751 with the fall of the Exarchate of Ravenna, or even 1130, when the Empire lost its last Italian possessions in Southern Italy[1]. The date of 476 is convenient to write on school exams, but its significance shouldn't be overstated.

[1] Personal note: If I were asked to chose one date for the end of the Western Roman Empire, gun to the head, I'd lean towards 751 because the fall of the Exarchate had all kind of consequences (most importantly the Pope deciding to throw his support behind the Frankish Carolingians), while 476 was not even the last time that the whole of Italy was under direct Imperial control (that'd be 572).