Icelandic historian Jón Viðar Sigurðsson gives three reasons for the saga-writing of the Icelandic aristocracy:

• To create and maintain social differences between chieftains (the aristocracy) and the rest of the population.
• To compete socially with other members of the aristocracy. Books were expensive, thus a symbol of power and status. Sigurðsson theorizes that Iceland has such a distinct literary tradition in part due to a lack of other status-seeking options. There were, for instance, no opportunities for glory in warfare in the second half of the 13th century and 14th century.
• Later (late 13th century and 14th century) saga-writing, in particular, was motivated by the desire of the Icelandic aristocracy to maintain/reconnect links with the Nordic countries (Iceland became more isolated during the same period, which was not good for such a peripheral entity, see the collapse of the Nordic settlement in Greenland when ties broke with Norway for example) by tracing the ancestry of Icelandic aristocrats to well-known kings and heroes that the contemporary Nordic kings could also trace their origins to.

See this source and this source.

I'm reading the sagas in English translation at the moment. It may or may not be helpful, but Robert Kellogg in his introduction mentions that along with authorship and some other factors, it's not entirely known. Surely it's a remarkable thing, not only that so much was composed in a short time by a civilization we would consider marginal, but that they largely did so in prose as opposed to verse. A major assumption is that Icelanders of the thirteenth and fourteenth centuries developed a collective urge to record tales about their forebears, who they were and what they represented, probably because they sensed that this legacy was in danger of being lost - the sagas are set centuries earlier and many record the old pagan ways being subsumed by Christianity. As to why the literary output flared up at that time, in that bulk, I don't think there's much consensus.

Not a historian, BA in Scandinavian Studies: Another factor would be that the christianisation of Iceland was comparatively peacful and fast. Literate christians (that had little to no trouble settling down and therefore saw little harm in existing pagan material) documented but also used and transformed an existing oral tradition that survives until today; whereas in other (scandinavian) countries, missionaries would feel the need to subdue the old ways in order to gain control.

A few notes regarding the question: At the time of the writing of the Icelandic sagas, Iceland was proportionally closer in population to other nordic countries. The population was roughly 1/3 of the population of today´s Norway. There also was a Norse population in Greenland, and the British Isles were Norse, meaning that Iceland was not as remote as implied. It should also be noted that it is not the existance or the size (around 11.000 pages) of the literary corpus that is unique, but rather that it is non-latin.

EDIT: I did not have sources for my origional claims of proportions, which came from memory. I stand corrected on the 1/3 - but non the less Iceland still at the time had a proportionally higher population compared to other nordic countries than today.