I visited Rome a few years back, and walking through the ruins of "Ancient Rome" and exploring the Colosseum were memories I'll carry through life.
I recently found myself wondering, at what point did it stop being Rome, so to speak, and start being what we think of as Ancient Rome now?
In this picture, you can see the modern city of Rome surrounding these ancient ruins. But at one point, I imagine they were just houses, business, government buildings, temples, etc. and no one thought of them as anything special. When did this change? What remains of the rest of Ancient Rome and what process did the city undergo to exchange historical ancient sites for modern urban buildings?
The simplest explanation is that the population of Rome declined massively after the fall of the Western empire. It had been the largest city in the world for centuries, with possibly over a million people (though estimates vary), and it collapsed from there to some tens of thousands- a few percent of the former population. Therefore, you have a huge number of buildings that are no longer being used, and are going to fall into disrepair and ruin. It wasn't until the nineteenth century that the population really began to recover, and by then all of the imperial Roman buildings that had survived were pretty well recognized as architectural relics.
Of course, it didn't take 1400 years for the buildings to reach their present state- I can't speak to precisely when it became popular to visit the ruins of Rome, but the first real heyday was probably when the Grand Tour became fashionable. This was basically a way for young men of means (primarily British, but also other Northern Europeans) to travel Europe and become cultured, as well as accumulate stashes of art treasures. This began in the mid-1600s, when Rome's population was still tiny, and a popular subject for paintings and other media of the era was contrasting the decayed grandeur of a Roman structure with contemporary Italian peasants going about their business- you can see a lot of this in Piranesi's Vedute di Roma series. So the ruins definitely were an attraction long before the modern city grew back up around them.
Short version: Everyone left and they didn't use the buildings anymore, so they got real old and run-down.
Maybe off topic but I think important to consider: Ancient Roman ruins were often pillaged and materials recycled for use in building all around Rome. Cheap marble dragged from the Coliseum and the Circus Maximus (to the extent that nothing really remains of it) was widespread during eras of massive building in the middle ages and renaissance. Furthermore sites remained in continual use. I'm sure you heard about the Easter observances in the Coliseum, for example. It isn't until recently that we've started to give a damn, as it were, about their historical significance. Even Mussolini demolished ruins in order to fulfill his egotistical ends. Very sad, in my opinion. Luckily most of the Forum Romanum remained buried.