For the Anglo-Saxons, the knowledge that it was the Romans whose city it was is less relevant than why those cities were there no more, or, well, why they were in such a poor state. However, for a direct answer, Gildas and Bede describe Britain as part of Rome (though possibly not with a specific statement that "The Romans built these") and this would have been understood by the educated people.
However, there was also an allegoric understanding where these came from and I wanted to focus on that.
Poetic Description
I recommend 'The Ruin', a splendid Anglo-Saxon poem. It features a lament on the state of the present as opposed to a glorious past, based on a township. The featured city is believed to be Bath.
... The halls of the city
once were bright: there were many bath-houses,
a lofty treasury of peaked roofs, many troop-roads, many mead-halls
filled with human-joys until Fate the mighty changed all that. ...
—'The Ruin'
Note I modified that links' "terrible chance" to "Fate the mighty" as in the Wikipedia translation. Wyrd (as in the original) had a specific meaning and "chance" doesn't really represent it.
Anglo-Saxon Worldview
Christian doctrine was primarily influenced by a few select works for the Anglo-Saxons.
Implicit in The Ruin is a philosophy of history, a way of looking at historical events—especially, an explanation for the fall and ruin of cities and empires.
—Doubleday, '"The Ruin": Structure and Theme'
Augustine, whose De civitate Dei was one of the most influential works for contemporary Anglo-Saxons, featured heavily and he was intent on explaining "the ways of God to man".
Orosius' Historiae adversus paganos tried to explain that "the prevalence of disasters in human history is the just punishment for sin, provided by a God who rules history and uses it to carry out His will".
Additionally, Gildas was able to construct history to fit what had happened:
In reproaching kings and clergy for their wickedness and threatening them with deserved punishment, Gildas draws on the Old Testament prophecies and on New Testament warnings. The history of the fall of Britain at the beginning of the work seems primarily intended as an exemplum for the sinners reproached in the latter part, showing them the end of their wickedness.
—Doubleday, '"The Ruin": Structure and Theme'
So the glorious Romans were thrown down because they were not fervent enough in their belief. The fall of a civilization was guaranteed by the people not giving thanks for their success, and vices and sin always abound in such times.
After the destruction, the desolate cities remain as monuments of the Lord's vengeance.
—Doubleday, '"The Ruin": Structure and Theme'
Reasons for the Fall
... Days of misfortune arrived—blows fell broadly—
death seized all those sword-stout men—their idol-fanes were laid waste—
the city-steads perished. Their maintaining multitudes fell to the earth.
For that the houses of red vaulting have drearied and shed their tiles,
these roofs of ringed wood. This place has sunk into ruin, been broken into heaps,
There once many men, glad-minded and gold-bright,
adorned in gleaming, proud and wine-flushed, shone in war-tackle;
There one could look upon treasure, upon silver, upon ornate jewelry,
upon prosperity, upon possession, upon precious stones,
upon the illustrious city of the broad realm.
Stone houses standing here, where a hot stream was cast
in a wide welling; a wall enfolding everything in its bright bosom,
where there were baths, heated at its heart. ...
—'The Ruin'
... implicit in these three descriptions are the three worldly loves that St. John counsels against: the lust of the flesh, the lust of the eyes, and the pride of life.
—Doubleday, '"The Ruin": Structure and Theme'
Baths were, of course, the cesspit of all sin, a place of fornication, and a promotion of luxuries. Bathing thrice a year was the hallmark of a saint while bathing plenty was only an indication that the people were avaricious, greedy, and lustful.
In other words, the city (and Rome in general) was ripe for falling because its inhabitants loved too much the worldly and not enough the otherworldly. The Lord was driven to smite the Roman (Britons) down as they had to be punished for their sin.