Reading this Wikipedia article makes it seem like the Gaelic world, and Ireland in particular, did a lot to spread Christianity to much of Western Europe. My question is why and how. It seems odd to me geographically that a culturally distinct island on the western extremes of Europe would have so much to do with spreading Christianity. Scotland and Ireland are both pretty isolated from the rest of Europe.
This is a big question that has no single answer, but I'll address two factors that I see as critical: (1) charismatic eremitism, i.e. holy men in the woods, and (2) monastic communities, each with their own saintly mascots and a commitment to proving their mascot's worth.
The importance of hermits is in some ways self evident. Hermits are people who go where no Christians have gone before. In terms of the physical spread of Christianity, it's part of a hermit's job description. In the early medieval world, men seeking an eremitic retreat moved along the highways of their age, which were the big maritime and riverine routes of northern Europe. Once they found an uninhabited spot, they set up camp. But these weren't the isolated spots in the wild that they sometimes pretended them to be—they were major traffic hubs. Columba set up Iona right in the midst of all traffic going between the North and Irish Seas, Fursey occupied an old Roman fort at the center of East Anglia's sea routes, and Cuthbert put himself right in the midst of Northumbria's maritime traffic. These regions went from having a minimal Christian presence to having a distinctly Christian place right at their very center.
With all the attention that these guys inevitably received, the most charismatic of them (i.e. the ones that we know about; the others were swiftly forgotten, leaving the misleading impression that all Irish-type missionaries were successful) attracted a band of followers. Some, like Fursey, eventually decided to move on, starting the process all over again, but others sunk in their roots and established monastic communities. Many of these guys soon had reputations that were bigger than life, and after their deaths, their communities were most likely to repeat the good stories and forget the bad. Before long, many of them realized that the revered founders of their monasteries had all the best qualities of legendary saints, and ought to be revered as saints as well.
There's a couple key things that a saint needs. First, a saint needs some sort of record of his or her life. Somebody had to write a book. Second, there needed to be at least one day a year when everybody got together and read that book. Usually, this was done on the anniversary of a saint's death. Third, a saint needed a place to be buried and have his or her relics preserved, and it's got to be nice. No self-respecting saint would allow his or her community to fall into disrepair. This means that the community had to be engaged in constant fundraising. This brings us to number four: a saint's got to have a good reputation. If reading about a saint's life on his or her feast day isn't bringing in enough donors, then the community needed new proofs that their saint hadn't abandoned them. Some wrote new accounts of miracles that had happened in their monasteries, proving that the saint still exercised power. Others 'translated' a saint's relics, which was basically a big parade that usually ended with the saint's bones getting placed in a new and better spot in the church.
It's important to remember that during all of this cult-of-the-saints kind of work, monks were also doing the duties of pastoral care. This was largely an age before parish priests, and monks seem to have taken care of a lot of the preaching, baptizing, and day-to-day work of spreading the faith. A number of them were also ordained so they could do priestly duties when needed, but this was long before the theology of sacraments was thoroughly hammered out, so there was less of a need for priests. (Burial, marriage, reconciliation, and even regular churchgoing all seem to have been negotiable.) Monks seem to have been able to minister to most people's Christian needs, and people could travel to the monastic church if they needed something particular.
Building and maintaining a monastic community didn't just mean attracting big land donations on those rare occasions when a saint's relics were brought out and rummaged through; it also meant drumming up more basic support like a wheel of cheese or an occasional cut of beef. And on very rare occasions, families might even donate a child to the church, perhaps hoping that this might provide the child with a better life than they themselves could provide while at the same time putting themselves in the good graces of God. Monastic pastoral care built up the relationships that made this continued support of manpower and material possible. This was Christianization at work in the countryside.
But this is how the process of Christianization went from Christian wilderness to monastic landscape. If the key to it is hermits infiltrating non-Christian places, then where did hermits come from? It's easy to imagine that young boys growing up in monasteries and reading about the lives of their saintly founders would soon be inspired to try similar heroic deeds. In fact, we have reports that this is what inspired Guthlac to go out and set himself up on the spot of the future Crowland Abbey, right in the marshes that joined East Anglia, Mercia, and Northumbria together.
But what about the very first Irish missionaries? That's a bit more of a problem. They might have benefited from certain Irish tendencies toward decentralized power, but it would be hard to identify anything resembling centralized power anywhere in Western Europe during the early age of Irish missionaries in the 600s, so I don't think that made Ireland all that unique. Perhaps as a more pastoral (cow herding) society, they encouraged a Christian spirituality that was a bit more transitory in the landscape. Perhaps also, early Irish hermits should be seen in light of a larger population of elite exiles who populated the early medieval north. Most of these exiles were political, but in the Irish case, some men started self imposing a religious exile as well. And this of course fits in with the broader mobility of medieval populations that scholars today are just beginning to appreciate. So there's lots of possible explanations, and it's definitely an area with a lot of possibilities for further research.
I'm not an expert in Ireland, but I've read fairly widely on the Christianization of Europe. For the material above, I've drawn on and would especially recommend:
Peter Brown, The Rise of Western Christendom, 3rd edn. (Oxford: Blackwell, 2013 [1996]).
Patrick J. Geary, Furta Sacra: Thefts of Relics in the Central Middle Ages (Princeton: Princeton University Press, 1978).
Ian Wood, The Missionary Life: Saints and the Evangelisation of Europe 400–1050 (London: Routledge, 2001).
While I can't answer your question directly, you got me interested to look up the history of Christianity in Ireland through the subreddit. I was able to come across this question which helped me understand how the system came to be.
u/depanneur's explains that the Irish political structure enthusiastically took hold of Christianity in an attempt to supplement their decentralized power, while at the same time Christianity in Ireland took on a much more distinct flavour than the more mainland variants. I found reading both the OP (who choose to delete their account) and u/depanneur's helped explain the environment that fostered the early adoption of Christianity in Ireland, and what the Irish were exporting.