The history is out there for those who wish to look. I have not re-checked what I vaguely remember but I think what I found was this. In summary the Argentinians abandoned the islands and by treaty agreed to accept a tiny part which they then abandoned again. Since 1833 the Islands have been English and the population has been English and they want to be English. The Argentinians tried an invasion and got flung out neck and crop (which in itself would have brought any legitimate claim, if there had been one, to an end).

The claim is profit-motivated colonialism. Much like the US claim to Cuba.