They found the islands empty and there's no evidence they were ever permanently inhabited before the European's arrival, though there are indications of transient visits by indigenous peoples from South America† ~1270–1400 AD. France then landed with a small settlement in 1764 (which they later surrendered. in '67, to Spain,) but the British, (unaware of the French colony) asserted sovereignty to secure whaling interests in 1765.

In the 1770's Spain & Britain contested for dominance. The English initially withdrew due to the American Revolution and the Spanish later abandoned their colonies 1811 due to internal economic & political upheavals; leaving the islands uninhabited again until 1833, when Britain reasserted sovereignty, though that is still contested by Argentina (who call the islands Malvinas, after the French Îles Malouines,††)

Today's population is predominantly of British descent with a deep connection to English culture; Red phone boxes, London-style buses, and pubs called “The Globe” or “Rose & Crown”, serving fish & chips, are common.

† Consistent with the Fuegian material culture; possibly the Yaghan.

†† From the Malouins, inhabitants of Saint-Malo in Brittany who comprised many of the colonists of a that 1st French settlement. - In 1982 Argentina staged a 74-day invasion / occupation of the islands prompting the Falklands War. Despite losing they still claim sovereignty, but, in a 2013 referendum, 99.8% (on a 92% turnout) voted to remain a UK territory.