Upvote:5
Paraphrasing liberally from the wiki article on Celtic Christianity, at least the following three factors played a role:
Because Ireland had not been part of the Roman Empire, the network of abbeys basically filled a bureaucratic vacuum. They ended up controlling large swaths of land following the growth of the monastic movement in the 6th century, and wielded great ecclesiastical and secular power.
Abbeys had very strong ties with the local ruling elites, in that abbots weren't necessarily ordained, and usually descended from one of the many Irish royal families. Further, the founding regulations of some abbeys specified that the abbotcy should if possible be kept within one family lineage.
Irish monasticism was notable for its permeability, in that people were able to move freely in and out of the monastic system at different points of life. Because of this, students would sometimes travel from faraway lands to pursue Latin scholarship in Irish monasteries. This, Ireland's isolation, and Ireland's lack of invasions during Early Middle Ages all fed into Irish monasteries becoming centers of Latin excellence.