Yes. The Anglo-Saxons/Early English were quite aware that the lands that they lived in were a part of the former Roman Empire, and the references to the "giants" are literary devices of the author.

In short, early Medieval England and the Anglo-Saxons as a whole were still engaged with the literary traditions of the western portions of the empire. Despite stereotypes and misunderstandings about the collapse of Roman civilization in much of Western Europe, it is not as if the people of these areas ceased to remember that the Roman empire had at one point ruled over them. Indeed the presence and trappings of Roman institutions remained an important cultural touchstone in Britain for centuries after the collapse of Roman life on the island.

We can see this in a variety of ways. Romanitas or the idea of aping the aspects of Roman life to confer legitimacy and power was a very prevalent idea in formerly Roman Western Europe. By embracing the trappings, symbols, and ways of the Romans leaders as diverse as the Ostrogoths in Italy, the Franks in Gaul, the Britons and Anglo-Saxons in Britain were able to draw connections between their rule and the days of Roman rule, which was multi suited. It gave them a greater prestige at home (and abroad) as well as provided for greater elements of continuity in the political landscape of Western Europe. This Roamnitas was expressed in a variety of ways, such as the passing and maintenance of legal frameworks from Roman times, the continued (attempted) maintenance of Roman public works and institutions, the adoption of Roman iconography and imagery on coinage and in artistic depictions, Britain was no stranger to any of these.

However where Britain is unique is in the degree of collapse that occurred. Tthe entire region, even extending into Gaul, was in a severe economic decline that predated the collapse of Roman power in western Europe. This made Britain both less attractive for Roman elites to defend and stay in, as well as a drain on badly needed manpower and financial resources. We can measure this economic downturn through a variety of different methods given the lack of historical literary sources that directly record it.

One aspect is evidenced by the abandonment of urban areas in Southern Britain, today England. Urban spaces in the Roman Empire needed vast economic support from both the surrounding countryside as well as reliance upon Imperial networks of trade to be sustained. Luxury goods that the urban elite of Britain relied upon such as pottery from Africa, olive oil from Italy, wine from Greece, and so on were no longer available in large quantities by the end of the 4th century and into the 5th. (Now they did not vanish entirely and there is evidence that these trade routes existed in some capacity for as long as a century or two after this period) This specialization of the Empire, allowing different regions to fill in the local resource shortages, was brought to a slow halt as the empire collapsed in the west. Unable to sustain themselves off of local resources the elite of Britain were faced with a few choices, loose status as Roman figures due to their lack of access to Roman urban life and forge a new living/identity as warlords, retreat to lands still held by Romans, or assimilate into the new populations that were cropping up in lowland areas of Britain.

The urban villa life of the Romanized elite, itself probably a small portion of the population though this is by no means universally accepted, died alongside Roman urban life and the former villa centers became estates for various warlords and newcomers, or were divided up into more manageable settlements. This was obviously quite disastrous for the people who owned these estates, and some clearly anticipated eventually returning as evidenced by coin caches discovered in many of these areas.

Other aspects of life changed too. Christianity famously disappeared from England during this time, only to be resurrected during the evangelizing movements of the 6th-7th centuries. Now while not all Christian communities vanished Christianity did take a back seat during this time and pagan traditions, not entirely removed from those popularized in Scandinavia, arose and came to dominate much of what became England. So this was one major change overall.

Other cultural aspects were slower to change and can only be seen with the benefit of hindsight. This includes changing cultural tastes in various aesthetic choices such as housing, jewelry, and clothing. Many new styles were formed that had their antecedents in both Roman, insular, and Scandinavian traditions. Over time however the cultural focus of much of lowland Britain shifted from the continent to Scandinavia (think the location and figures that dominate Old English epic poetry such as Beowulf, and this would not be undone for centuries, it took the Norman conquest to firmly shake England from the Scandinavia world.

did the established Anglo-Saxon kingdoms really begin to believe that the ruins were the works of giants?

One of the biggest problems in historiography is an apparent reluctance on the part of many historians to treat people in the past like, well, people; as writers capable of rhetorical devices, emotion, imagery or poetic licence. The Early English were a people who seem to have put great stock in words, whether written or recited, and who produced wonderfully evocative poetry. A relatively common theme in many surviving Early English texts is a kind of bitter-sweet rumination on the passing of time and the eventual fleeting nature of all things. It appears throughout Beowulf, for example, in the shattered glory of Hrođgar's hall, or Beowulf's eventual battle with the dragon. It's a theme in poems throughout the Exeter Book, and it's the theme, broadly speaking, of The Ruin, a rumination on the faded legacy of the once-great Rome, a 10th Century Ozymandias, if you will.

The physical legacy of Rome was still very present for much of the Early Medieval period in England, even if much had declined, decayed and vanished, and the pursuit of Romanitas - a sense of political, cultural and social "Roman-ness" - was a major concern for many Early English rulers, particularly after the Augustinian Mission of 597. Romanitas could be a kind of symbolic shorthand, an aspirational aping of Roman styles and glory to claim legitimacy and greater standing. The pursuit of Romanitas could appear in many forms. For example, to celebrate an alliance with Ceolwulf II of Mercia in the 870s, Alfred of Wessex minted an issue of pennies at London known as the "Two Emperors" series which is based on the Roman soldii of Valentinian II and Theodosius. A few decades later, his daughter Æthelflæd celebrated the Mercian restoration of Chester's Roman walls and its church by minting a series of pennies with pictoral designs that bear a similarity in particular to coins issued by the Emperor Nero with similar architectural motifs. These are deliberate, conscious copies, clear departures from the otherwise fairly distinctive and consistant designs of English coinage which had developed in the eighth and ninth centuries.

This is just one small angle; Early England was still litered with Roman construction which was often still in use or integrated into wider infrastructure. At Bath, for example, where The Ruin is postulated to be set, the early 10th Century saw the development of what is today known as King Edward's Bath amid the site of the previous Roman bathhouse, while the bath's large halls at least in part became used by the city's traders and guilds. Roman defensive circuits were re-used throughout the ninth and tenth century Burghal network, at sites including London, Rochester, Exeter, Rocester, Chester, Leintwardine and Cynuit. Roman roads were a major element of transport infrastructure, and indeed standard rent obligations for landowners by the late 9th Century - the Trimoda necessitas - included the upkeep of the network of Roman roads. Martin Carver even suggests that Æthelflæd's design of Mercian burhs, in particular at Stafford and perhaps at Eddisbury, were consciously based on lessons she had taken on Roman military organisation.

The legacy of Rome is also present throughout English writing: Gildas' de Excidio Britanniae clearly informs Bede's Historia Ecclesiastica, which in turn forms the basis of the contemporary narrative of Late Antiquity seen in sources such as the Anglo-Saxon Chronicle and, indeed, The Ruin. The ninth century Medical textbook known as Bald's Leechbook references a swath of Classical authors, including Pliny, Galen, Celsus and Oribasius. What's actually quite interesting about The Ruin is the very English way in which its author imagines the social life of the eponymous site:

...where at one time many a warrior, joyous and ornamented with gold-bright splendour, proud and flushed with wine shone in war-trappings; looked at treasure, at silver, at precious stones, at wealth, at prosperity, at jewellery, at this bright castle of a broad kingdom.