The way I understand it is that Christianity became a huge religion because it was dominant in Europe and it was dominant in Europe because after the fall of the Western Roman empire, barbarian tribes would adopt Christianity. But from what I know Christianity at that time wasn't a very powerful or major force.. Bearing that in mind I thought that it is quite weird for strong barbarian tribes (like the Huns and Vandals) to become commited to this religion. I asked my history teacher about this and he told me that the Christian church was very well organized (having systemized ranks and hierarchy, like Bishops, the Pope, etc.) and that there was a basis of Christianity followers which made it easy for tribes to rule over a larger population. Are these the main reasons or is there anything more?
I think there is a simple misunderstanding of chronology here that we can quickly clear up.
Christianity was already dominant by the fall of the Empire, and many of the barbarians were already Christian when they entered the Empire.
The fall is debated - many people will argue the Empire never fell, merely transitioned into a different form in Late Antiquity - but it's traditionally dated to 476, when Romulus Augustulus abdicated.
The Empire is generally considered largely Christian since Constantine (d. 337), and Christians had legal toleration since the Edict of Galerius in 311.
Meanwhile Ulfila (c. 350) had been teaching the Goths and through them various other Germanic people the Arian flavor of Christianity that the Council of Nicea had found heretical in 325. When Alaric sacked Rome in 410, he was an Arian Christian.
So it's not a matter of a pagan Rome collapsing, barbarians moving in, and then converting. The Empire was already Christian and the barbarians were a different flavor of Christian, or a lot of them anyway. (Also, Arianism wasn't entirely expunged from the Empire, but I'm not equipped to discuss all the theology and who was what; maybe there's other help around here).
That leaves unanswered your question, why barbarians decided to convert to a flavor of Christianity generally judged a heresy. But you can ask it in a better way: Why, before Christian Rome fell, before Christian barbarians entered the Empire, did those barbarians convert to Christianity?
I can’t speak for the specific peoples you mention, but I can provide some insight into how the pagans in Denmark and the surrounding areas, over a longer period of time, eventually became what one would consider christian. The majority of this information comes from the work of Adam of Bremen, who wrote about the history of the archbishops of Hamburg/Bremen (Deeds of Bishops of the Hamburg Church) from 1070. Whether or not his analysis is a reliable and objective source is up for debate, but generally it is considered to be accurate. Excuse my english as I am not a native speaker and also by no means an expert in this subject.
Adam described a general four step method in which the Hamburg Church’s mission was accomplished in and around Denmark:
With regards to the question of power, and the ability to force the barbarians into this new system, keep in mind that the church worked with the king of Denmark (Svend Estridsen in 1070) at the time of Adam of Bremen in order to promote this new order. Of course it started over a hundred years before that under Harald Bluetooth. Also, seemingly no one had the will or means to create a fleet and an imperium the size of the one held by King Canute in the earlier part of the 11th century, so the scope of the later christian mission was relatively small in scale at first. They didn’t initially try to enforce tithe in the whole of Denmark, as that would be impossible. I also don’t believe that all the conversions necessarily happened violently, at least not among the peasants. The actual vikings, which were the elite, were the ones suffering from this newfound relationship between king and church, and there was a lot of violence here. The church was not happy with the culture of raiding and pillaging and it sought to put things in order through tithe, among other things.
Getting the pagans to pay tithe proved to be a tremendous effort and it was a lengthy process, and it wasn't until the 12th century under King Niels that it was actually widely introduced to the danes. Adam wrote that the things that were supposed to be free of charge, like baptisms, confirmations and marriage ceremonies, were expensive because the danes didn’t pay their tithe. They simply had not learned the ’proper’ way to do it. Initially, the tithe was enforced through some of the viking elite, and that proved to be a very expensive arrangement because these supposed public servants would sometimes just rob the people instead of doing their job. Eventually the king and church figured out that they should task officials with the job of collecting the tithe instead. Tithe was steady income and it was in a sense the looting of the people by the king and church, instead of the pillaging and raiding of other peoples by the vikings. At least that is how it could have seemed from the perspective of a viking warrior, and it is understandable how they would view this as hostile to them and their way of life. At this point, even if a lot of the people eventually started to pay tithe, you probably wouldn’t consider them christians. As stated it took a long time for people to actually believe in God and many probably held on to their previous pagan gods until they died, even if the official symbols were now crosses instead of Thor's hammer, for example. Later the common people who did not believe in the pagan gods were still highly polytheistic and instead of maybe believing in God, they worshipped saints. It took hundreds of years for the common folk to convert to monotheism. At the rituals it wasn't really a problem that the people imagined one thing and the priest another.
To the question of why they adopted christianity in the first place, remember that the church and christianity brought with it new technology, writing and education, agricultural reforms, and all in all a more stable society where raiding and pillaging wasn’t necessary in the same scope that it previously was at the time of the vikings. Over course there is a pretty big overlap where the vikings were christian and still raiding, but this was the direction everything was heading. Of course tithe wasn’t fair to the people at all, but in the long run and on the grand scale the cooperation between the church and the king marked the beginning of an actual state and I see it as an overall positive thing in terms of the future of the country and compatibility with the christian neighbours.
Sorry if this isn’t what you are looking for, I just have a hard time helping myself. Feel free to ask questions.
Sources:
Deeds of Bishops of the Hamburg Church
Danmarkshistorien i Grundtræk
edit: corrected some things after reading it through again.