The short answer: sheep. According to the earliest records of the settlement of Iceland, the island was forested everywhere between the highlands and the coast when the Norse first arrived. Often, these semi-historical accounts in the mediaeval sources have to be taken with a grain of salt, but this assessment has been backed up by modern science, which estimates that approximately 40% of the island was covered by birch forests prior to settlement.
Over time, the settlers cut down trees for charcoal, tools, houses, and ships. Because Iceland’s environment is relatively harsh, once trees were felled in large numbers, it was difficult for them to grow back.
Perhaps the largest impediment to reforestation, however, was sheep grazing. It has long been traditional in Iceland for farmers to let their sheep roam in highland pastures during the summer, and then to collect them in the fall. This sheep grazing caused immense damage to Icelandic forests, from which they are still recovering. To this day, most tree plantations in Iceland need to be fenced in, to prevent sheep from destroying young saplings.