Like everything, it gets complicated.
At one time, Britain was involved in a world wide land grab - Empire building if you like. At the time, it was something people believed in, that untamed lands should come under the "wise rule" of Europe.
Despite what some of the apologists would like us to believe, the arrival of the Europeans must have been a welcome relief, as especially in Africa, the local rulers were incredibly nasty.
Of course, the whole empire idea ended, as throughout history as soon as people become "comfortable" and by comparison well off, they want to rule themselves - or an elite wants a piece of the action. Empires shouldn't have happened on reflection, the rule wasn't always benign, but that's history.
So back to the Falklands, this was an area claimed by Britain - presumably to secure southern sea routes. The British were there before Argentina existed....important point.
Argentina decided later they fancied these islands, as they were so near presumably. They have been arguing for them ever since.
Britain probably doesn't really want them, but after the Argentine invasion in the 80's, it's near politically impossible to ever think of giving them away. I've heard also that the locals aren't the loyal British people the media might like you to think, but are.....let's say not exactly friendly.
Another little known thing - neither Britain or Argentina wants to take the question to an international neutral arbitrator. Reason being, the decision could go either way - Argentina knows full well their claim is pretty shaky indeed.
Another little known thing, Britain helped build Argentina in the first place, and was one of Britains oldest friends. Rather a pity things have turned so sour over a bit of rock.
Bet it's all down to oil