Well, Vinland was first mentioned (in vague terms) in Adam of Bremen's late-11th century chronicle over the diocese of Bremen-Hamburg, and that's the closest source in time to the actual Vinland journeys. The Vinland Sagas from Iceland which were written down centuries later and were not known outside Iceland until Scandinavian humanists of the 17th century started taking an interest.

At that time, the historian's aim was most often to find past glories for his people and country to revel in. So interpreting Vinland as the discovery of North America was immediate. (and by the standards of the time, practically a believable and humble claim compared to - for instance - Olof Rudbeck trying to prove Sweden had in fact been the lost Atlantis)

The rest of Europe was not terribly interested in Sagas at the time. Spurious claims to pre-Columbian discoveries of the New World had also been popping up since not long after the discovery. (e.g. the Zeno family's "Estotiland") So for instance, Benjamin Franklin relayed in a letter (July 7, 1773) having heard about the claims from [Finland-]Swedish professor of botany Pehr Kalm:

You have made the most of your argument, to prove that America might be known to the ancients. there is another discovery of it claimed by the Norwegians [..] About twenty-five years since, Professor Kalm, a learned Swede, was with us in Pennsylvania. He contended, that America was discovered by their northern people, long before the time of Columbus; which I doubting, he drew up and gave me some time after, a note of those discoveries [...] The circumstances give the account a great appearance of authenticity. And if one may judge by the description of the winter, the country they visited should be southward of new England, supposing no change since that time of the climate. But if it be true, as Krantz, I think, and some other historians tell us, that old Greenland, once inhabited and populous, is now rendered uninhabited by ice, it should seem that the almost perpetual northern winter had gained ground to the southward; and if so, perhaps more northern countries might anciently have had vines, than can bear them in these days.

This would be fairly representative of that era; when learned Scandinavians (including non-historians) had a basic familiarity with Saga literature by that point in time. They'd been studied and translated and published in modern Swedish and Danish. But they were not well known outside the region. Yet.

That would change in the 19th century when England (and other countries) were struck by a trend for romanticized vikings (as part of the greater Romanticism trend). Much of the whole 'viking' concept and interest that continues to this day stems from that period. For most intents and purposes that's when the vikings were invented. Sagas were adapted and translated and people projected thier own time and ideas onto the vikings they created for themselves. Hardly anyone in the English speaking world had heard of a 'viking' in 1800 but by 1900 it was as known as now, and people had the fully formed image of the horned helmets and 'viking funerals' that persists to this day despite lacking any historical basis.

Anyway, so in that century the Vinland sagas also became common knowledge, and many people believed them. This can't really be termed a 'realization' though. There was no hard evidence supporting the idea of Scandinavians in the New World, so the entire thing hinged on the credibility of the sagas. But the Vinland Sagas were written centuries after the fact and were full of fantastical elements like one-footed monsters. Even in the 19th century, when the Sagas and Eddas were generally presumed to be older than thought to be now, and read less critically than now, there were skeptics. But those were matched with claims more far-fetched than anything that'd be made today; such as the Danish historian Rafn claiming in the 1830s that the ruins of an old mill in Rhode Island were from a Nordic round church.

I've written in greater detail on the historiography here. The short of it is, that starting with Fridtjof Nansen's 1911 book In Northern Mists and more generally with Weibull's saga criticism, Scandinavian historians became far more skeptical of Sagas in general and not least the Vinland Sagas.

So, ultimately it was the discovery of site at L'Anse-aux-Meadows in 1960 that really elevated the idea of a Norse visitors to the New World to fact.