The immense growth of Christianity in times of polytheism is baffling as it is, but really I wonder how they were able to convince rulers such as King Olaf to abandon their pagan beliefs and violently enforce Christianity on their subjects.
Success begets success.
Let's go back to Constantine for a second. Once he was sort of converted (but not baptized) he promoted Christians at court. Which made people want to be Christian, at least nominally, so they could be promoted. His successors (except Julian) actively suppressed and fought against paganism. But the mere fact that the Roman emperor was Christian was enough to make many people want to convert to be closer to him and gain his favor. This was true of many Germanic tribal leaders (who then forced it on his subjects) as well as ordinary bureaucrats.
And that never stopped being true. For early Germanic rulers, such as Clovis I of the Franks, having the Christian connection meant the possibility of being connected to other Christian monarchs (for instance his wife was a Christian princess). You could make more marriage alliances, you had a connection. Once the Norse carved out real kingdoms they wanted to be taken seriously as any monarch would. It was beneficial to make marriage alliances (especially by the time of the Viking conversion no Christian king will marry his daughter to a pagan).
But that's just part of it. Christianity also had the benefit of organization and an automatic bureaucrat class. Most of your subjects in Norway aren't going to be literate. And they aren't even going to be able to think in terms of administration (the Norse were incredibly intelligent, crafty and mercantile people I am not suggesting they were incapable but it wasn't their way). It just wasn't part of the tradition. But the entire set-up of the Catholic Church is based on the organization of the Roman empire. And the Romans were nothing if not efficient.
Christians move in and suddenly your country is divided into dioceses and archdioceses. Which also serve as effective units for taxation and record keeping. Clergy also served as key figures in the administration of law. In much of medieval Europe they served as Chancellors. Lower down the chain you had literate people who could transcribe. Even lower down the chain you had people in most towns and all dioceses that could spread a message given from up high.
If you are a pagan king who has just formed a kingdom out of what had been a bunch of warring minor kings and a traditionally independent people you want the already built in administration the Church has to offer you.
Another feature is the very heart of paganism compared to Christianity (or Islam, for that matter). Pagans could easily believe in their own gods but also believe in your gods. "Sure, this Christ guy is alright. We'll throw him in the pantheon next to Thor." But if you are a Christian then there is only one God and everyone else is wrong. So, a pagan is fine marrying off his daughter or son to someone who has different gods. So what? You worship your gods and I'll worship mine. A dedicated Christian will be more reluctant to do so (not that it never happened). A dedicated Christian monarch will be less willing to deal with you, in theory (although in practice people do what they have to for their political ambition). The basic connections that made up power alliances are suddenly strained. And they wouldn't have to be.
On top of all of that, Christians were good at it because the Christians were often smart about it, at least at first. The Popes were usually pretty practical. Let the natives worship both Jesus and Thor for a little while. Ween them off Thor. Be patient. Let them migrate their native mythology and traditions into Christianity (this is why so many pagan traditions were ultimately adopted into Christianity). This is how England was converted. By the time of the Norse conversion they didn't have to be that pragmatic because the kings were willing to use force. And the Norse were very passionate about their gods so that was probably necessary.
To sum up, Christians became powerful and efficient converters because they were close to power and everyone wants to be close to power. They also had administration to offer. And, beyond all of that, in the early days they were usually pragmatic about how they went about it.
First of, i can only write about the christianisation of Europe, especially of Slavs, the Pruss and Scandinavians in 7-10th century, but the pattern of conversion was not much different.
You ask how were they so successful at it, and the answer is, they were NOT. The Church was simply sufficiently focused on expansion of its faith, and send missionaries over and over and over again until it stuck (as exampled by the christianisation of Pomerania, which took several centuries, and countless efforts by the Church and christian feudal rulers of Poland who had sent many missionaries to their doom at the hands and spears of the pagans).
As mentioned above, the Church was deeply interwined with the feudal power structures. For a pagan ruler, it was a choice of either converting to christianity along with his retainers and family (and thus, symbolically, his tribe/nation) or be destroyed/subjugated by the powerful christian kingdoms. It was not uncommon for a land (like the medieval Masovia) to have converted rulers, and pagan commoners for centuries, without much of a conflict. In fact, "pagan" comes from latin "pagus" which means "villager" which highlights the difference between the converted elites and old faith of the commoners.
Besides faith itself, the missionaries brought with them basically everything the european civilisation had in store; literacy, modernised agriculture, advanced architecture, and awesome (in the traditional sense of the word) religious art, rite and culture. The very building of a church by itself was a powerful tool to inspire awe, since most pagan religions that far away from the Mediterrean shores had no temples. The combination of those factors made a powerful impression on the "barbarians".
Pagan religions (by which I mean non-abrahamic faiths from Europe, excluding the ones from Asia) had a very weak structure compared to the Church. Aside from the druids and vates of the Celtic faiths (which failed long before the Church rose to its full power), Europe had no organised "priesthood" that we know of, and what we think of as monoliths of "Norse faith" or "slavic religion" were actually very balkanized and syncretic network of various shamans, priests and "secular" folk (for example, the patrarchs of the family among the Pruss) performing the rituals. This made forming any kind of strong opposition against christianity difficoult, and when it happened, it was more for political reasons that religious ones (like the famous "Pagan Revolt" in Poland in 1032 and another one in Hungary in 1046, both of which were politically motivated smokescreen for a power-grab).
Sources:
S. Gavrilovych "The triumph of Peter and Paul. Brief history of the Catholic Church" Warsaw 1990 A, Andrievska "The culture and beliefs of the Pruss" Lodz 1987 B. Sondermann "the Northern Crusades. A brief summary" Oslo 1966 Christiansen, Erik The Northern Crusades. London 1997.