For example, the French missionaries in the early 1600s managed to unseat the local native American religion and instead convert them in to Christianity, despite the native American beliefs being thousands of years old. How did they go about doing this?
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 asked: 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.
The Kongolese, for example, willingly converted themselves because their mythology spoke of a people of white ghosts who were very advanced and would bring the Kongolese many gifts and knowledge. Then the Portuguese came and sort of fulfilled that prophecy, I guess.
My source for this is a website in Portuguese, who does list its sources, I don't know what the protocol for these kinds of situations is. http://www.paratiando.com/negroreino.html
Islam also converted very fast and effectively, through merchants and colonists who formed families with the locals and by kingdoms who accepted Islam as a way of achieving political allies, prestige and a solid legal code.
I have been watching and thinking about this thread for a while, and I think the problem is you have framed a general question ("How did Christian missionaries succeed?") on the presumption that there is some general answer. I would consider it deeply dishonest to try and answer that question. What one ought to do is consider the methods and efficacy of specific Christian mission efforts in specific times and places. One might be able to extrapolate after considering several such historical studies, but only by showing that the factors involved were comparable.
In your question text you give a specific situation. Is 'French missionaries in the early 17th century in North America' what you want to know about? Or is it an example only?