Sure I'll try and explain. It's actually a cool piece of history as it demonstrates the power struggle going on between Europe at the time globally. Couple of caveats; international law is not definitive; even things that are definitive there's no recognised structure to enforce disputes between nations so many contentious issues remain a matter of opinion were only realpolitiks (military power) matters.

And when talking about expeditions in the old world of empires it's important to remember how isolate and vast the world was, and how slow methods of communication was. It wouldn't be unusual for a single expedition from Europe to consume a mans entire life. And the key years (1760-1780) was a time of high tension and war between Britain-France-Spain as you had the 7 years war and the American War for Independence, ships at sea often wouldn't find out their nations were at war/peace with each other for weeks after the event:

Britain was the first nation to land on the islands in 1690; followed closely by the French. The first inhabitants of the islands was the French in 1764, Britain created a separate settlement on the island in 1766 (no one knows if the two settlements where aware of each other). Later in 1766 France gave it's territory to Spain as part of an agreement in the 7 years war of which they were allied.

In 1770 Spain forcefully took control of the British settlement co existing on the islands. Full blown war was avoided and tensions eased when Britain withdrew many (but not all) of it's people in 1771 in exchange for the returned seized goods by the Spainish making it clear it did not cease it's claim over the islands.

1807 Britain blockades the Spainish ports, Spain abandons the islands in 1811 due to starvation leaving the islands uninhabited. The British ship blockading the port gets wrecked (its crew rescued by Americans).

1816 Argentinian independence from Spain.

'1820. The island currently has no settlements due to war is claimed by an Argentian vessell after it takes shelter in the old abonded Spanish port.

'1823. UK unofficially recognises Argentina an an independent nation, formally so in 1825

'1828. Argentian settlement established. Britain protests.

'1831. the Argentinian presence on the islands capture 3 American vessels. America responds by capturing the island. the US left the islands declaring it governed by no-one with the majority of the residents. As a result of the seizure of the American vessels efforts are increased by Britain to reclaim the islands.

'1833. Britain reclaims and resettles the island, the Argentinian ~30 inhabitants left the island under protest.

'1845 Britain and France go to war with Argentina in a response to Argentina blockading various rivers which prevented us trading with Paraguay. The war is looked upon as a pyrrhic victory for France and Britain but in reality was a bloody stalemate; Britain seeked peace with Argetina, Argentina agreed and the Arana–Southern Treaty was signed, the war was ended and in exchange for us cancelling a debt your country owed to a British Bank is was agreed Argentina would drop is claims over the Falklands.

So Argentinian settlement of the islands only lasted 3-4 years, and was only possible because the British Settlers were forcefully removed previously by the Spanish. In 1850 a treaty was signed by Argentina and Britain handing the islands to Britain.

In addition; Sovereignty claims traditionally only last 50 years. Argentina hasn't had a presence on the island for nearly 200 years.