Why are the Falkland Islands British?

Because the people who live there (and are ancestors of the original and only populations) have chosen to remain British.

Simple no?

Might makes right. Since islands are not connected physically to countries, a strong navy decides who owns an island.

The old, old story. Exploration and competition between nations to acquire new territory. The British first occupied West Falkland in the 1760's but were quickly repelled by the Spanish who had previously acquired it from the Spanish. For the next few years the Spanish and British wrangled over it and the Spanish occupied East Falkland. When the British left West Falkland they did not renounce their claim upon it.

Meantime, when the Buenos Aries government become independent of Spain itmade its own claim on the islands.

The rest is pretty much well known history.

the once royal britan navy once ruled the seas and the british empire was vast

The British got there first and the people and even the sheep on the island are British.

The Falkland Islands had no indigenous population prior to their settlement by our ancestors– the Islands were entirely unoccupied. First claimed by Britain in 1765, the British, French and Spanish periodically had garrisons in the Islands until 1811, when all the garrisons were withdrawn.

https://www.youtube.com/watch?v=s1VQPqo01wc&t=10s