Thanks for the shout out /u/y_sengaku!
I'll post the answer here in its entirety to save you all a click
Norse religion is an oxymoron. There was no canon, no dogma, no clerical hierarchy, no organization structure, no infrastructure to support priests or sacred sites, and no popular participation in the pagan "religion". Instead I think it is far more apt to describe "Norse religion" as a religious tradition, which I will shorthand to just Norse paganism. Now this is ultimately somewhat Eurocentric of me, with "religion" bearing the hallmarks of Western religions like Christianity, however since your question is concerned with Christianity specifically lets just roll with it.
I'm going to split this answer into a few parts, one detailing Norse paganism, one Christianity in the early Middle Ages, and finally the process of conversion.
Part 1: Norse Religious Tradition, what it was and what it was not
Norse mythology is something that many of us in the western world are broadly familiar with, but only on the surface level. Odin is the All-father, Thor has a hammer, he fights giants, Loki is in there, and so on. However what we "know" about Norse mythology is mostly derived from a series of saga stories that were written down by Christians, and mostly one particular Christian (Snorri Sturluson) in Iceland centuries after conversion. The deities that we know and love, Heimdall, Tyr, Loki, all of whom are actually relatively unattested in archaeological evidence are common in the sagas, and vice versa, deities such as Ullr rarely appear in the saga literature despite far more evidence of a widespread cult based on place names. How are we to reconcile this difference between the literary evidence and the archaeological, especially in light of the reliability of the literary evidence compared to the archeological?
There are a few other written sources that are slightly more contemporary, such as the Poetic Edda (which predates the official conversion of most of the Norse world, but only just) and Saxo Grammaticus's Gesta Danorum which was written by a Danish Christian. Ibn Fadlan's account of the Rus people in his own journeys is also often used as a source on Scandinavia, despite the fact that he was writing about Russia and modern scholarship is increasingly nuancing the idea of Scandinavian domination of Russia.
To be clear though, using these sources to try and reconstruct the cosmology, theology, eschatology, beliefs, practices, rituals, and view point of Norse pagans is a fool's errand. The sagas have about as much to do with the practice of Norse paganism as Disney's Hercules does with Graeco-Roman paganism of the 4th century BC.
So with that out of the way what do we know about Norse paganism and what are our sources? (In the interest of time and space, I'm not going to be detailing each individual practice, ritual, and so on that we have evidence for, but rather detail a broad conclusions that some scholars have arrived at)
We are largely left with archaeological evidence (physical objects such as rune stones, artifacts, place name evidence, and so on), contemporary accounts from outside the Norse world, and extremely curated selections from the surviving corpus of Old Norse literature. So what do these sources tell us? What secrets can they reveal to the intrepid researchers of today?
In short, that the old Norse pagan religious tradition was elitist and extremely insular (not to mention barbaric, including human sacrifices and, if Ibn Fadlan is to be believed, the ritualized gang rape of slaves) with little popular participation and little buy in beyond the nobility. Norse paganism was hardly a core aspect of Norse "heritage" if the rapid and successful conversion to Christianity is a useful metric to go by. Indeed the religion likely varied extremely among the vast majority of the population and the paganism practiced in one part of Scandinavia likely bore little relation to that practiced in another. Evidence from across the Norse world shows that there was a great deal of variation in practices such as burial (cremation vs inhumation) and local cult popularity (as evidenced by the wide variety in theophoric place names).
The charismatic aspects of the religious tradition, veneration of Odin, ship burial/cremation, Valhalla, were probably the exclusive domain of the aristocratic elite of the Norse world. The average Norse person would not have been a participant in the same religious life as the elite of society. The average farmer, trader, slave, who lived in the Norse world almost certainly did not share the same conception of their own religious tradition as the elites of Norse society did. What good would Valhalla be to a farmer after all? Instead their worship likely focused around less well known deities with far less ostentatious displays of piety and worship.
Indeed it seems that the religion, such as it was, was incredibly tied to elite participation for legitimacy and practices. Elites in society, such as, but not limited to the King and his immediate family, were the ones who were keeping the religious practices going with ostentatious sacrifices including humans, horses, and other goods and food items and celebrated the deities and figures of the religion in their own oral traditions that would eventually be recorded by the same strata of elite members of society after conversion. They were also the ones who patronized the oral tradition of skaldic poetry that was eventually recorded by Snorri. Without elite buy in, the Norse pagan tradition could not, and eventually did not, maintain itself.
As Anders Andren says about the religion to sum up what I have covered:
Instead the religion must be regarded as a series of partly overlapping traditions, differing from place to place and from time to time, and also between different age groups, sexes, and social groups. Perhaps the shared Scandinavian features, such as boat graves and sacral place-names, should primarily be viewed as the religious expressions confined to an aristocracy with wide-ranging connections all over Scandinavia.
Part 2: Christianity in the Early Middle Ages
At the onset of the Viking Age, loosely defined as 800-1100, Christianity had completed its dominance of Western Europe and was starting to creep east. The former Roman lands of Italy, Gaul, Britain, and Iberia had all been converted (or reconverted in the case of England) by this time, and the Roman Emperor Charlemagne had started to spread Christianity at the tip of a sword to the Saxons, various English missionaries arrived in Germania (and Scandinavia), the Slavs were starting to convert, and the Roman church was starting to take a shape into a more familiar form to modern people.
However there were still some critical differences. Modern practices such as clerical celibacy, private confession, widespread access of communion, and so on were still some time off. However, Christianity had several things going for it at this time that made it stand out among the competing religions and traditions of early Medieval Europe, chief most among these were prestige and infrastructure.
Christianity at this point was the religion of the Roman Empire, indeed two of them. The Eastern Roman Empire had been Christian for centuries by this point, and the newly crowned Roman Emperor in Aachen, Charlemagne, made Church reform a high priority of his own. This association with the most powerful realms in Europe made Christianity appealing as a prestigious good that could be given.
One of the most important aspects of Christianity is of course baptism, and it was a powerful tool in the arsenal of conversion. Baptism, and the subsequent creation of God-Father/God-Son relationships was a powerful means of creating cohesion and loyalty in Early Medieval societies.
Christianity was also the gateway to greater trade opportunities, centralization, and infrastructure.
Trading was often restricted, or attempted at least, between Christians and non-Christians, and many luxurious trade goods such as wine and Frankish jewelry (popular in pagan Anglo-Saxon England for example) were appealing to non-Christian populations. However of more direct import especially to would be convert kings, were the benefits that Christianity brought to a ruler's administration and efforts to centralize authority. Latin literacy was a pre-requisite for the administration of medieval kingdoms (despite the presence of the vernacular in both Ecclesiastical and Secular literature in places such as England), and Latin literacy came through the Church. Furthermore a king who embraced Christianity could offer a more prestigious religion to his followers (mediated through baptism) that also brought alongside it greater connections, such as trade, to the powerful realms in Western and Southern Europe.
Finally, even at this early stage, Christianity was a more popular religion, and I mean that in the sense it appealed to the populace at large. As I pointed out above, popular participation in Norse paganism was limited, but this was not necessarily the case for Christianity. While weekly masses in the vernacular were still some ways off for the majority of the population, many parts of Western Europe were more directly engaged in religious practice (and not necessarily in a way that benefited them, I'm sure the peasants who worked on monastic land were not necessarily thrilled to be doing God's work) in a way that pagans in Scandinavia were not.
While more can always be said, /u/Steelcan909's answer to Why did the norse gods die out so quickly? in this subreddit should be referred to at first.
(Edited): fixes partly broken link.
As u/Steelcan909 mentions there was simply not enough incentive to keep the old faiths though I would like to go a bit further into why the powers that were could have been incentivised to convert in the first place. As with a lot of Early Medieval Scandinavian History it is difficult to know exactly why things progressed as they did since very few contemporary sources have survived until today.
What we know for certain is:
This could indicate that the reason, at least partially, that Harald I converted was to minimize the chance of a crusade lead by The Empire. If this indeed were the case, the transfer of his father Gorm from his burial mound to the church might have been an assurance to the clergy that he was sincere in his belief.
Other things worth mentioning in the case of Danish conversion to Chritianity are:
That some medieval scholars claim that Harald's son Sweyn I revolted against his father as a direct cause of the conversion to Christianity, which could indicate that the Danish elite was not christened before Harald and that some of them even opposed the idea.
The Danes continued to raid and invade other Christian nations well after it officially became Christian and thus were supposed to stop indiscriminately attacking other Christian Realms. Swein II attacked William the Conqueror both in 1069 and 1074/75.
There have been archeological finds such as a stone used for metalcasting which has carvings for casting amulets of both crosses and Thor's hammers,^(3) which could mean that there was a significant amount of time where Christianity and the old faith coexisted.
A lot of not only Danish but Scandinavian folklore continued,^(4) or even continues,^(5) to be believed to this day as superstitions which most scholars agree have their roots in pre-Christian beliefs.