Argentina, Chile, the UK, France, Norway, Australia and New Zealand each claim some, but not all, of Antarctica. No state at all claims Marie Byrd land, and there is also an unclaimed territory of uncertain area around what would otherwise be the Norwegian claim's corner at the South Pole. Some of the territorial claims even overlap.
Why did these states select the claims that they did? Why didn't the UK (which was the first state to assert a claim) claim the entire continent?
The UK at one point did intend to exert sovereignty over the entire continent. There are letters written to the Governors General of Australia and New Zealand expressing a goal of bringing the entire continent in the British Empire. Why the British interest? At the time there was a lot of money to be made in whaling. And by exerting control over the land, the British were claiming the right to regulate and tax whaling in the Antarctic Sea.
The French and Norwegian claims were made in direct response to that. The French claim was made along a section of coast discovered by the French explorer Jules d'Urville. The Norwegian claim is likewise due to Norwegian discovery of Peter I Island and the adjoining coast. The Australian and New Zealand claims are part of British claims that had been governed by them and thus went with them as they gained full independence. All five of these countries have treaties with one another recognizing these claims.
The Chilean and Argentine claims are not recognized by anyone else. But they do date to the same period (early 20th century). Both countries have made their claims based on territorial proximity.
The unclaimed area remained unclaimed because it was more isolated. It had no adjacent territory. There were no nearby whaling stations to exert control over. So any claim there would have been less supportable. And there was no significant value to be gained in claiming the territory.
Mainly because at first there was very little reason to take a continent that seemed to have nothing on it apart from snow and ice.
Later the Antarctic Treaty System was created in 1959 to prevent the more people from claiming it and to prevent the Cold War militarising it. Given its position far away from them both American and the USSR was willing to agree.
Of course all of this was done when Antarctica was of purely scientific study, know that oil has been discovered in the are who knows what might happen.