This question alone and the attempt to answer it sums up almost all the research involved in late antiquity/early medieval studies.

A lot of it depending upon a close examination/deconstruction of what the words "Rome" and "fell" mean.

What is Rome? What is Roman? What does the word "Rome" mean to someone in a certain geography and a certain time span, and how does that meaning change when you switch to someone else in another geography and time span?

What is a "fall?" How fast or how destructive must something be for it to be counted as a fall? How fast or destructive was Rome's fall? What if it wasn't a fall but a change? What's the difference between a fall and a change?

You're likely not going to get a satisfactory answer, because the simplest questions to ask actually tend to be the hardest questions to answer.

But we do know quite a bit about the process. I would recommend, if you're interested in knowing more, begin to refine your question down further, down the whirlpool into a more specific time, geography, and thought.

While what Juvenalis said is ultimately right, there still is a reason, why earlier historic approaches came to the depiction of Rome's fall, so i'd like to answer the second part of your question by outlining the development during Late Antiquity. Bear in mind this is an approach to show what lead to the conclusion of "Rome's fall" based on examples. History isn't static until a single event happens and brings change, resulting in a new static episode of history, like older historic research (looking at you 19th century) sometimes implied by it's interpretations of historic sources. It's rather an ongoing development. Because of that you have to take a look at the development during Late Antiquity to answer your question what the "fall of Rome" means or what it "actually looked like" (to really know what it looked/felt like, you would have to go back in time).

One important aspect in Late Antiquity is the successive separation of the importance of Rome as a place and Rome as an idea. The idea of Rome for example still was very important to inhabitants of the Roman Empire, when the emperor already resided elsewhere (e.g. Treves or Ravenna) as it can be seen reading Ausonius or Apuleius. Even in the middle ages the importance of Rome as an idea can still be seen in the title of the king of the Holy Roman Empire.

The emperor leaving the city of Rome is one aspect showing the loss of it's importance. Another process during Late Antiquity is the change in society and army. Over a large period of time Rome's army consisted of a core of Roman soldiers (though they were accompanied by a greater number of socies and "foreign" auxiliary). That however changed after the Barbarian Invasion and the Hun's invasion. From that time on there were armies with a core of "barbarian" forces serving the Roman Empire. Eventually that lead to a shift of power, again successively not in a single moment, from Rome and it's emperor to the barbarian foederati and their leaders.

Another important development of Late Antiquity is the split of the Roman Empire. Rome was no longer the single center of the entire empire as there then were two, Rome and Constantinople/Byzantium. Successivley the eastern half of the empire became more and more important and likewise its centre.

These are just examples of development, but combining them you can already see Rome fading away and losing its importance.

As for sources i used German titles(especially König, Ingemar. Die Spätantike. Darmstadt 2007), so i don't know if they're of great help for you, but you can read the development of Late Antiquity in any introductionary work to the matter.

It's just a narrative contraction for a complex series of theorised events which occurred over decades. Historiographical fashions change with time; the fashion used to be to bang on endlessly about 'rise and fall of civilisations', which is thankfully not so much the case today, but you still get people who like to come up with these dramatic and irritating generalisations.

Source: Philosophy of History, in the Stanford Philosophical Encyclopedia.