It's silly to claim that all Britons hate the Argentines. But they did go to war with them, and there are still some pretty strong feelings held by SOME Britons about that. It's important to note that Argentina didn't invade the Falklands because they really and truly felt compelled to claim them for themselves - they went in because Argentina was going through a very difficult time economically, and the sitting government thought that going to war would distract from that situation. They were very wrong in thinking so.
Secondly, the Argentine government mistakenly believed that Britain would not opt to defend the islands with full force, which was obviously not the case. Many Britons died defending the islands and many died in the fight to reclaim them. Public opinion on the war in Argentina was divided with some people - mostly ultra-nationalists supporting the invasion, but many Argentines never wanted to go to war with Britain, and the issues that led to the fighting were only exacerbated by the invasion and subsequent conflict, so in the end, the few Argentines who were for it began to see things differently.
Though located much closer to Argentina than to Britain, the islands have never belonged to Argentina. The Argentines never had a legitimate claim to them, and even if the islands had belonged to Argentina at some point in the past, the vast majority of the population consider themselves to be British and feel very strongly about remaining part of the UK, so what Argentines might think is irrelevant. Going to war with Argentina was mostly seen as righteous in the UK as Britain was defending its territory and its people, and as happens with any war, the enemy aggressor was viewed unfavorably and a lot of that has yet to disappear. Perhaps in the decades to come the animosity that some Britons hold for Argentina will fade until it completely disappears, but because the war occurred relatively recently, many people in both countries continue to have strong feelings about it.