How did Constantine suddenly flip the stigmatized, cult-looking Christianity as the dominant religion in the Roman Empire without being persecuted? And why?

Because by the time of Constantine's conversion, Christianity was no longer an obscure cult made up of subversive elements from the lower classes, but was firmly entrenched among the class of urban professionals and rising new military and bureaucratic officials that made up a very influential chunk of the Empire's demographics.

The Crisis of the Third Century brought substantial social and cultural changes to the Roman Empire. Most notably, it brought a rising tide of "new men" from outside the traditional upper classes of the empire to prominence. Their avenue to power was primarily through the military, for the Crisis was a series of divisive and devastating civil wars between self-proclaimed emperors:

For the Roman Empire was saved by a military revolution. Seldom has a society set about cutting out the dead wood in its upper classes with such determination. The senatorial aristocracy was excluded from military commands in about 260. The aristocrats had to make way for professional soldiers who had risen from the ranks. These professionals recast the Roman army.

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The soldiers and officers [who fought in the Danubian campaigns], who had seemed so raw to the Mediterranean aristocrats of a previous age, emerged as heroes of the imperial recovery of the late third and early fourth centuries...The army was an artesian well of talent. By the end of the third century, its officers and administrators had ousted the traditional aristocracy from control of the empire.

These "new men" formed the basis of a new imperial bureaucratic and military administration that would preside over a recovery that spanned the fourth century. Their rise heralded the dawn of a new system of advancement that relied more on merit than birth. As a result, men from disparate regions, cultures, ethnic groups, and religions could rise to high positions with the administration.

This new culture and influx of talent allowed for men with Christian beliefs to quickly entrench themselves into the highest levels of Roman governance once Constantine converted to Christianity.

The reign of Constantine, especially the period from 324-337, saw the final establishment of a new "aristocracy of service" at the top of Roman society...After the conversion of Constantine in 312, the emperors and the majority of their courtiers were Christians. The ease with which Christianity gained control of the upper classes of the Roman empire in the fourth century was due to the revolution that had placed the imperial court at the centre of a society of "new" men, who found it comparatively easy to abandon conservative beliefs in favour of the new faith of their masters.

So now the question is how Christianity was so appealing to this wave of "new men" (outside of how conversion allowed them to rise more quickly in the court of a Christian emperor).

Christianity offered a few distinct advantages compared to other religions at the time, chiefly its culture of community, exclusivity, and egalitarianism. Anyone could become a Christian no matter their ethnic, economic, or former religious background. And once you were a Christian, you were part of an exclusive community, of which many were men from well-off economic backgrounds and invested their wealth in improving that community. Thus, Christianity appealed to men who felt they lacked a social identity, and/or were trying to carve out a new niche for themselves in post-Crisis Roman society; and since the turmoil of the Crisis uprooted many people and produced a new group of ambitious, talented social risers, Christianity found itself with a wealth of new converts.

The Church was also professedly egalitarian. A group in which there was 'neither slave nor free' might strike an aristocrat as utopian, or subversive. Yet in an age when the barriers separating the successful freedman from the declasse senator were increasingly unreal, a religious group could take the final step of ignoring them. In Rome the Christian community of the early third century was a p[lace where just such anomalies were gathered and tolerated: the Church included a powerful freedman chamberlain of the emperor; its bishop was the former slave of that freedman; it was protected by the emperor's mistress, and patronized by noble ladies.

For men whose confusions came partly from no longer feeling embedded in their home environment, the Christian Church offered a drastic experiment in social living...

The Christian Church suddenly came to appeal to men who felt deserted. At a time of inflation, the Christians invested large sums of liquid capital in people; at a time of increased brutality, the courage of Christian martyrs was impressive; during public emergencies, such as plague or rioting, the Christian clergy were shown to be the only united group in town, able to look after the burial of the dead and to organize food-supplies...Plainly, to be a Christian in 250 brought more protection from one's fellows than to be a civis romanus.

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What marked the Christian Church off, and added to its appeal, was the ferociously inward-looking quality of life...the wealth of the community returned to the members of the community alone, as part of the "loving-kindness of God to His special people.

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The appeal of Christianity still lay in its radical sense of community: it absorbed people because the individual could drop from a wide impersonal world into a miniature community, whose demands and relations were explicit.

Once Christians gained access to the highest levels of government via the "new men", and those "new men" carved out their own position among the elite classes of the Roman Empire, Christianity continued the process of adapting to the new culture of the classical world. The Crisis of the Third Century had brought more than civil war - foreign powers hostile to the Empire, such as Sassanid Persia and the Germanic tribes along the Rhine, had taken advantage of the weakness of Roman borders and launched raids and invasions into imperial territory. The mood of the apparent collapse of the "civilized", classical world took deep hold across the Roman Empire, and the narrative of Christianity was well-suited to adapt to this new mood:

Hence the most crucial development of these centuries: the definitive splitting-off of the "demons" as active forces of evil, against whom men had to pit themselves. The sharp smell of an invisible battle hung over the religious and intellectual life of the Late Antique man...To men increasingly pre-occupied with the problem of evil, the Christian attitude to the demons offered an answer designed to relieve nameless anxiety: they focused this anxiety on the demons and at the same time offered a remedy for it. The devil was given vast but strictly-mapped powers. He was an all-embracing agent of evil in the human race; but he had been defeated by Christ and could be held in check by Christ's human agents.

The early fourth century was the great age of the Christian Apologists...They claimed that Christianity was the sole guarantee of [classical] civilization - that the best traditions of classical philosophy and the high standards of classical ethics could be steeled against barbarism only through being confirmed by the Christian revelation; and that the beleaguered Roman empire was saved from destruction only by the protection of the Christian God.

When Constantine very publicly converted to Christianity, he was inundated by a flood of Christian "new men" who desired his patronage either for their own advancement within the government or for the advancement of their community's interests under his rule. By surrounding himself with Christians, Constantine surrounded himself with Christian propaganda, and allowed that propaganda to spread throughout the empire. And because Christianity was already entrenched among the urban middle class, combined with the eastern empire (the focus of Constantine's power and attention) being considerably more urbanized and developed than the western empire, this led to the majority of the entire empire becoming firmly Christian from the bottom up, despite the resistance of the traditionalist pagan aristocracy:

This prolonged exposure to Christian propaganda was the true "conversion" of Constantine. It began on a modest scale when he controlled only the under-Christianized western provinces; but it reached its peak after 324, when the densely Christianized Christianized territories of Asia Minor were united to his empire.

Constantine's nephew, Julian the Apostate, who became emperor after the death of Constantine's son Constantius II, was a firm pagan who sought to roll back Christian infiltration within the upper levels of Roman government. But his premature death on the battlefield in 363, only three years into his reign, smothered those plans in the crib. The new Christian domination of the Roman world was here to stay.

Source: The World of Late Antiquity, by Peter Brown (pub. 1971)

I encourage you to seek out further replies and sources to this question. My sole source is a secondary one, and an old one, despite being an extremely influential work in the historiography of the late Roman Empire.