First off, it is of course impossible for us to talk about "the ancient Romans" in the context of this question. Are we talking about people from the city of Rome in A.D. 30? Provincials from Syria in the second century? Gallo-Romans of the fourth century? Are we talking about ordinary working people, about equestrians, about the senatorial upper class? It is somewhat like asking if "Americans like(d) President Reagan". The Roman Empire was extremely long-lasting, extremely vast and very multi-cultural, it accommodated a wide range of different peoples with different cultures and different mentalities and changed massively over time as well. With that in mind, I will try tackling your question.
This is a topic that has been debated quite controversially, and it’s not settled yet. Unfortunately, there is just no way for us to holistically reconstruct the mentalities of ancient people. Too vast is the gulf of time separating them from us, too fragmentary are our sources, and the vast majority of people who lived and died back then do not get a voice at all, for they left us nothing. We can however make speculations based on what has come down to us in the form of inscriptions, archaeological remains and literary sources.
The “divinization” ie. deification of a monarch had first been introduced to the Roman Empire in 42 B.C., with the post-mortem consecratio of Iulius Caesar, through the process of which he was turned into an ascended state deity. After Augustus it became standard political practice for emperors to be declared divus ("divine", but also used as synonym for "god" or something inbetween god and human) after their death by their successor, because it was a very convenient way of legitimizing the successor as divinely ordained as well as passing judgement on the previous emperor posthumously. Indeed, it wasn’t just that monarchs were deified after their deaths (apotheosis): the Roman emperors went to great lengths to acquire sacred honors and surround themselves with an aura of divine power while alive, too. On its face, this must’ve run quite contrary to the traditional Roman state religion, as in that of the people of the city of Rome before the Principate. Appian has summed up the Romans’ somewhat contradictory relationship to their monarchs:
"There an altar was first erected, but now there stands the temple of Cæsar himself, as he was deemed worthy of divine honors; for Octavius, his son by adoption, who took the name of Cæsar, and, following in the footsteps of the latter in political matters, greatly strengthened the government founded by Cæsar, and which remains to this day, decreed divine honors to his father. From this example the Romans now pay like honors to each emperor at his death if he has not reigned in a tyrannical manner or made himself odious, although at first they could not bear to call them kings while living."
- App. Civ. 2,148.
Roman state religion did not allow for statements about the nature or genealogy of gods, the relationship between gods and humans was regulated in cult practice, nothing in this conception allowed for a “bridge” between the divine and human realm. Neither did the Republican constitutional system of rule allow for a sacralization of the ruler, or the client patron system allow for ritual veneration of the patron. So where then did this come from?
Well, the Hellenistic east of the empire saw all this quite differently. Here is where the Imperial Cult originated. It was well-acquainted with cultic worship of a monarch and immediately applied the idea of the savior-god (sōtḗr) and divine benefactor (euergétēs) to the Roman conquerors. The provincial council (koinon) of Asia already honoured Caesar that way:
The cities and peoples and tribes in Asia honour Gaius Julius Caesar, son of Gaius, the high priest and imperator and consul for a second time, the manifest god descended from Ares and Aphrodite, and the common saviour (sōtḗr) of humankind."
- Ephesos, 48 B.C. (Sylloge Inscriptionum Graecarum: 760)
The same of course happened to Octavian (later Augustus), and he was happy to receive these honours: thus, the koina of the Hellenistic East continued in their task of cultivating the ruler’s cult, as they had done under the monarchies before them. This way, the political interests of the city-states were directly connected to maintaining the worship of the Emperor. In the East, where the Greeks had lived under monarchies with divinized rulers for centuries, and the other peoples for millennia, the Roman monarch with his universal pretensions could only be imagined as an agent of higher powers, equipped with divine consecrations. This went so far that Emperor Vespasian was asked to perform wonders for the populace of Alexandria, to heal sick and blind people, for the god Serapis had told them to seek out the emperor. Tacitus mentions that Vespasian weighed the possible political effectiveness of his acts in the East before working his supposed miracles (Tac. Hist. 4,81).
In the provinces of the West (Gaul, Spain, Sardinia and Sicily, Africa, Germania and later Britain), the preconditions were different. The gods were usually seen as more distant, associated with the forces of nature instead of human deeds, there was no “hero-cult” of mythical humans with direct (often familial) connections to the otherworld. Still, however, the ruling classes were conscious that they would have to give their loyalty and thanks to Augustus for restoring peace. Augustus recognized the stabilizing effect of cultic acts of loyalty and established provincial councils in the West, tasked with maintaining the imperial cult, to channel the general approval into worship and bind the ruling classes to the person of the emperor. Under Vespasian, this system was expanded to all provinces of the West, not just the less pacified ones, to give the provinces a direct outlet to air grievances as well as continually renew their expressions of loyalty to emperor and Imperium.
Besides the provincial councils, all cities of the Imperium as well as private persons and associations autonomously funded a large variety of ritual honors for the emperor. The imperial centre did not interfere: the temples, altars, columned halls, and games impressively showed the good will of the cities and citizens to concentrate the divine powers of the imperial house on themselves. By the middle of the first century, the Imperial Cult was inexorably tied to being a loyal part of the imperium.
Now, of course, this obvious break in Roman tradition could not be made too overt. The worship was conducted in a way that made no concrete statements about the Emperor’s personal holiness. How close a living emperor’s relationship with the gods actually was, was always left ambiguous. It is important to mention here that public cult rituals, though overtly “religious”, did not have to be tied to the other-worldly beliefs of any ancient person. A Christian subject of Rome could publicly sacrifice to the honor of the caesar augustus to show their deference to the Imperium and its ruler, while worshipping their own god in their community. There was no neat separation of the religious and the political in the ancient world. The Imperial Cult probably had a fairly minor impact on the personal religious feelings of the people. To this day, archaeology has not brought to light any artifact in the private homes of any imperial subject that could prove that the emperor was in fact venerated as a personal god. This public “religion of loyalty” evidently required no personal piety.
In conclusion, we can say that the Imperial Cult was at its core a civic, public, ritualized form of showing and reaffirming loyalty to the emperor and the Imperium expressed through familiar religious/political rituals. In some way, it “democratized” honoring the emperor and made him less dependent on the senatorial elite, for unlike the bestowal of traditional Republican offices and titles, revering the emperor in public was open to all his subjects. It was an instrument of the imperial machinery of Rome with great political relevance and the apotheosis was a significant posthumous honor, but it remains contentious if a significant number of Romans and/or their subjects considered the emperor an object of personal veneration such as their own gods.
The Imperial Cult mostly lost its significance as a legitimizing factor during the turmoils of the third century, when basically all the traditional pillars of imperial legitimacy such as senatorial approval, Republican titles and offices, sacred honors and consent of the people of the city of Rome broke away in favor of acclamation by the military alone, with everything else an afterthought. What survived of it through the third century vanished with the Constantinian shift and the Christianization of the Empire in the fourth century. Roman emperors would instead look to the Christian church to sustain and uphold their rule.
There is actually a very good and exhaustive book on this subject.
In a nutshell, it's a difference in shades of grey (how traditional roman religion operated) vs. the binary (how Christianity operated). The problem is that almost all our modern conceptions of religion revolve around an Abrahamic/Christian binary framework, so there is no room for "middle divinity," only "God" or "not God."
From the mentioned book:
"...heavenly honours could in antiquity also be accorded to mortal men, and this fact raises a fundamental question: what distinction did the man–god divide in these rituals actually signify or reflect?
The difference is one of nature or, for lack of a better word, zoology: God is the sole example of another ‘species’, radically different. That is how we instinctively tend to interpret the man–god divide of Graeco-Roman divine cults. Yet the phenomenon of ruler cult in antiquity—and elsewhere outside monotheistic cultures, for that matter—shows that this interpretation is at best inadequate.There is, however, another option: the man–god divide in the pagan context could also be taken to reflect a distinction in status between the respective beings, rather than a distinction between their respective natures, or ‘species’. That is the model which I will test in this book: divinity as a relative rather than as an absolute category." (p.26)
And another choice passage:
"A crude parallel may further understanding. For instance, antonyms such as ‘large’ versus ‘small’ are relative terms which can be taken, in strict principle, to correspond to ‘divine’ versus ‘human’ in Roman pagan terms... To an ant a mouse is large, but to a cow it is small. This does not mean that either the ant or the cow is wrong, or that their distinctions are blurred, confused, or ambiguous. It simply means that there are no absolute criteria to determine what is large and what is small. Unlike monotheistic cultures, pagan antiquity had no absolute criteria by which to determine divinity, nor had it any clergy or holy texts to expound or set dogmas, and thus the only real dogma was tradition itself." (p.31)
Combine this with the fact that Roman religion did not have an extensive theology, and was primarily a religion of "practice" rather than belief. You sacrificed to the imperial cult because it was the right thing to do, not necessarily because there was a contractual and reciprocal effect. That reciprocity sometimes occurred is not the question, but to demand it would be to wander into the realms of "magic", because in all honesty, religion/magic and law were on the same spectrum: attempts to bind by those under, attempts to not be bound by those above.