There is archealogical evidence of ancient human habitation on the islands but nothing for many centuries before the French established a military outposts there in the 18th century, followed in short order by the British who set up whaling stations with the own defensive military garrisons. Essentially at the time of British and French claims to the islands they were uninhabited and unclaimed.
The French did try to establish a civilian settlement there by importing a farming community, but it wasn't very successful. The French farmers were not familiar with the terrain or the climate and struggled to survive. It was too expensive for the French government to keep this fragile community supplied and it was repatriated. When the French left the British took sole control of the islands and declared them British sovereign territory which the French who were the only other nation with a legitimate claim, did not dispute, and to their credit never have to this day.
The British supplemented their seal and whale oil industry with hill farmers from Wales and Scotland who were more familiar with the conditions than the French had been. Supported by substantial whaling revenues they managed to develop a thriving wool industry.
This all happened before Argentina was founded as a country. Their claim rests on the outlandish idea that every territory in south America once ruled by the Spanish should be Argentina's by right as they are imperial Spain's descendants in the region. The trouble is, lots of other south American countries could make the same claim. Argentina asserts that the Falkland islanders are an "implanted population introduced by colonialism" conveniently overlooking that their false claim to the islands by definition makes white Argentinians themselves a colonialist implanted population. By their own logic they should return Argentina to its indigenous people.
.... just for clarity my lineage is from yous fuckers having to own everything 
