From an earlier answer on medieval Latin European perceptions of Rome:

Textually, we have an incredibly rich trove of accounts of visitors to Rome. On one hand, the collection is perhaps not quite as interesting as we might want: the Venn diagram of "people who were both literate and whose writings are likely to have survived" and "people with an awareness of a basic history of the Roman Empire and its decline" is basically the first circle inside the second, especially from the mid-11th century on. On the other hand, their shared knowledge of and appreciation for ancient Rome offers a good basis for comparison of different perspectives.

Benjamin of Tudela is a good place to start for an important reason: in the face of Rome's role at the heart of medieval Christianity, Benjamin was Jewish! He came from Navarre in Iberia, and his meandering travel account catalogues the Jewish communities he traveled among around the Mediterranean. You can read his full account of Rome and Roman Jews here (Cntl/Cmd+F for "Rome" is easiest if the link target doesn't work), but to excerpt a few bits:

There are many wonderful structures in the city, different from any others in the world. Including both its inhabited and ruined parts, Rome is about twenty-four miles in circumference. In the midst thereof there are eighty palaces belonging to eighty kings who lived there, each called Imperator, commencing from King Tarquinius down to Nero and Tiberius, who lived at the time of Jesus the Nazarene, ending with Pepin, who freed the land of Sepharad from Islam, and was father of Charlemagne.

There is a palace outside Rome (said to be of Titus). The Consul and his 300 Senators treated him with disfavour, because he failed to take Jerusalem till after three years, though they had bidden him to capture it within two.

In Rome is also the palace of Vespasianus, a great and very strong building; also the Colosseum...There were battles fought here in olden times, and in the palace more than 100,000 men were slain, and there their bones remain piled up to the present day. The king caused to be engraved a representation of the battle and of the forces on either side facing one another, both warriors and horses, all in marble, to exhibit to the world the war of the days of old.

In Rome there is a cave which runs underground, and catacombs...In the church of St. John in the Lateran there are two bronze columns taken from the Temple, the handiwork of King Solomon, each column being engraved "Solomon the son of David." The Jews of Rome told me that every year upon the 9th of Ab they found the columns exuding moisture like water.

Although Benjamin observes that some of Rome is standing/inhabited and some is ruins, he does not distinguish which is which in his description (nor does that distinction allow for, as we will see, inhabited ruins). However, he is keenly aware of the history of the ancient Roman buildings and those who lived in them. Those stories--what Rome was--matter more than what they are. He takes note of great buildings, natural features, and smaller monuments. I also think the detail about the columns of the Temple seized and appropriated into a Christian church are fascinating and significant. Especially in recounting the miracle story of the local Jewish community, Benjamin shows that Rome could have a sacred geography for non-Christians--something I, at least, am not used to thinking of.

Notably absent from Benjamin's record, on the other hand, is commentary on the fall of Rome. For this, believe it or not, we have to turn to Christian writers. In their stylings, a very real admiration for classical antiquity aligns with the medieval Christian theology of history that saw a "world grown old," decaying towards apocalypse and only ever renewable by God. 11th-12th century cleric Hildebert of Lavardin, eventually archbishop of Tours, wrote two famous poems de Roma which both celebrate and mourn the ancient city as he found it at the very end of the 11th century. Here's an excerpt from one:

The city now is fallen; I can find

No worthier epitaph than “this was Rome.”

Yet neither the flight of years, nor flame nor sword

Could fully wipe away its loveliness

[…] Bring wealth, new marble, and the help of gods

Let craftsmen’s hands be active in their work—

Yet shall these standing walls no equal find,

Nor can these ruins even be restored

The care of men once built so great a Rome

The care of gods could not dissolve its stones

Divinities admire their faces carved,

And wish themselves the equal of these forms

Nature could not make gods as fair of face

As man created images of gods

With Hildebert, praises of Rome move into a more emotional register, but also a more intellectual one rather than practical/geographical. His words are grounded in ancient Rome's buildings and especially its art but he evokes the splendor of a lost civilization rather than the immediate materiality of buildings rooted in history. It's also significant that Hildebert's praise, while overtly of the artistic qualities of ancient Roman art, is actually directed at the human artists. He elevates the abilities of humans of old especially compared to present ones, whose skills and vision could never possibly measure up.

A few years later, the English traveler known as Master Gregory famously followed Hildebert's footsteps to Rome. Gregory actually knew one of Hildebert's poems--he quotes it in his own little travel guide-like account!--but takes his commentary a step further.

The sight of the whole city is, I think, most wonderful, where there is such a multitude of towers, so great a number of palaces, as none can count. When I first saw the city from far off, I was overwhelmed and remembered Caesar's view of it, when having conquered the Gauls and crossed the Alps, he exclaimed substantial quote from Caesar...

This beauty passing understanding I long admired, and I thanked God who...yet has magnified there the works of man with immeasurable beauty. For even if Rome falls into complete ruin, nothing that is intact can be compared with it. As has been said [by Hildebert, in fact]:

Nothing can equal Rome, Rome even in ruins

Your ruins speak aloud your former greatness

The ruin of Rome shows clearly, I think, that all temporal things are near their end, especially when the worldly center of all things, Rome, daily languishes and decays.

Rome as the "worldly" center of the word is one of those little noteworthy turns of phrase. In medieval Christian sacred geography, the center of the world was Jerusalem. Here, though, Gregory focuses on the human component in Christian world/salvation history--and he is even more explicit than Hildebert about the decline of the present from earlier greatness.

We're used to a "decline and fall of Rome" narrative as Christians supposedly ruining the great rationality/progress/technology of pagan/philosophical Rome. Medieval Christians actually took part in the view of a Roman golden age compared to their own; for them, however, the rise of Christianity was less a cause of decline than an inevitable step towards ultimate divine redemption.

Gregory relates one more detail I want to highlight here: he tells us that many of the statues from the days of pagan Roman glory were dismantled by (very important) Pope Gregory I! We can agree on one hand it's quite noteable that he's repeating a story about Christians actively opposing the preservation of pagan art, and not very approvingly. On the other, this Gregory projects the 'desecration' onto another Gregory several centuries in the past.

In fact, the appropriation and remixing of ancient Rome into a Christian city was ongoing throughout the Middle Ages. Even in the later 15th century, with "Renaissance" adulation of classical antiquity building to a fever pitch, prelates in Rome were still plunder the Colosseum and deserted palaces for stones for their own lavish building projects!

This brings us to the last thing I want to talk about: archaeological evidence for "what people thought" of Roman ruins, evidence that perhaps helps us get beyond the view of the absolute elite of the elite of high medieval society. The Colosseum is the famous example here, since it enjoyed many afterlives throughout the Middle Ages. Most famously, it eventually became a little neighborhood for artisans! Quarrymen and blacksmiths set up residence, even building the occasional shop for horseshoes and other goods. Eventually, a monastery was constructed in and around part of it. And all the while, tantalizing blocks of stone were usurped for building projects elsewhere.

Visitors to Rome saw "ruins" and "desertion," and the ashes of of past splendor. People who lived in Rome may well have seen that, too. But they also saw promise for the present and the future: what could be out of what had been.