How did we find out that vikings settled in North America? Do we know why the settlement was abandoned? How big it was? Where exactly it was? And all that.
We know about the norse exploration of North America from "Eiríks saga rauða" (Eirik the reds saga) and "Grænlendinga saga" (The Greenlanders saga). They divide North America into three areas from north to south; Helluland (Land of flat stones), Markland (Land of forests), and Vinland (disputed, but possibly land of grapes, land of meadows, or land of luck/happiness). The location is usually thought to be eastern Canada, where Helluland is Baffin Island, Markland the Labrador coast, and Vinland usually thought to be Newfoundland, but the dispute over the meaning of the name in Old Norse has caused some uncertainty in terms of the extent of Vinland (According to Adam of Bremen, Vinland comes from grapes, thus where grapes could've been found during the 1000s affects the location).
From what we can tell from the sagas at least one settlement was abandoned due to hostilities between the settlers and the indigenous people, or the "skrælling" as the saga authors call them, and it seems only Vinland was ever attempted colonized. However expeditions by the settlers on Greenland seems to have continued at least until the 1300s, as Icelandic annals mentions logging expeditions to Markland.
Sources:
Lectures by Lars Ivar Hansen at the University of Tromsø
Lectures by Paul Oddvar Liseth at Tyrifjord videregående skole
"Grænlendinga saga"
"Eiríks saga rauða"
"Leiv Eiriksson, Helge Ingstad og Vinland, Kjelder og tradisjonar" (2001) written by Jan Ragner Hagland
"Norsk historie 750 - 1537" (2011) written by Ole Georg Moseng
Reports of Norse travels to North America survive in two sagas that were copied into Icelandic manuscripts, sometime during the later Middle Ages. Apparently the Greenland Norse were sufficiently impressed with these expeditions to preserve stories about them, whereas only archaeological traces survive of their expeditions further north around Baffin Bay. However, by the time these stories were written down, many details had become confused or forgotten, and the two sagas differ so much that specialists think they tell us almost nothing of certain value regarding the Norse presence in Vinland.
Nevertheless, a pair of archaeologists in the mid-1900s hoped they could find enough clues in the texts to direct them to a Norse settlement. They traveled the coasts of Canada, comparing the landscape to the saga accounts, and by an extraordinary combination of good luck and perseverance, they found archaeological remains at a place called L'Anse aux Meadows at the northern tip of Newfoundland. Their discovery is the only evidence that Viking Age Norse peoples visited North America that scholars accept.
These excavations there turned up three longhouses dating to around 1000 and built in Icelandic and Greenlandic styles, together with a handful of artifacts that left no doubt that this was a Norse and not an Amerindian site. The buildings seem well suited for three ships crews, and certainly no more than 90 people. There are no indications that this was actually a "settlement." Most tellingly, there were no traces of women or livestock. (In contrast, the sagas tell of at least one woman traveling to Vinland and giving birth.)
Archaeologists concluded that this site must have been associated with the Vinland stories, but instead of being a permanent settlement, it seems like temporary lodging and work facilities. This accords well with both sagas, which indicate that Vinland was explored for profit rather than migration. Greenlanders would sail to L'Anse aux Meadows in early summer, use it as a base for gathering resources and perhaps trading with Amerindians further south, overwinter at L'Anse aux Meadows, then return to Greenland when the seas became passable the following year. The relative lack of archaeological finds suggests that the crews were well disciplined as they collected their things to leave.
It seems that after an short period of exploration, Greenlanders decided to abandon their visits to Vinland. The sagas suggest that this was due to hostility with Amerindians, but at least at L'Anse aux Meadows, this was not the case. The departure was disciplined, not under duress; there are no signs of attack or combat (such as arrowheads or graves); and there are no signs that Amerindians even lived in the area at that time. It is possible that the Norse had hostile contact with Amerindians along the coasts sailing toward the Saint Lawrence estuary, but it's even more probably that the Norse came into conflict for resources with Amerindians in Baffin Bay. Some of these stories might have worked their way into the story about Vinland or been exaggerated for effect, long before they were recorded as sagas in Iceland.
It's more likely that they deemed travels to Vinland too dangerous for its resources. Ninety men would have represented 2% of the estimated 4500 Greenlanders and the three ships would been invaluable to the Greenland communities, since they had no local wood with which to build ships of their own. A trip to Vinland would represent an extraordinary risk, where a single storm could wipe out a proportion of the community and its wealth comparable to the devastation of the US Civil War.
The resources of Vinland were simply not worth this risk. North American timber, which was probably the most precious resource, could be acquired without traveling so far south. Baffin Bay actually holds a wealth of resources, as Amerindians have proven through several millennia of continuous occupation. And the Norse on Greenland seem to have adapted fairly well, surviving there for centuries even as the climate got colder and contacts with Iceland and Europe became increasingly tenuous.
James H. Barrett, ed., Contact, Continuity, and Collapse: The Norse Colonization of the North Atlantic (Turnhout, Belgium: Brepols, 2003).
Stefan Brink and Neil Price, eds., The Viking World (London: Routledge, 2008).
Edit: clarity