In all practical likelihood, these claims have had no affect since the 1961 Antarctica Treaty which was signed by each of the nations claiming rights to land in Antarctica.
The motivation is simple, if oil or other valuable natural resource were to be discovered in Antarctica in an area claimed by one of these countries, they would claim sole right to the profits from these resources. However, the legitimacy of these claims would need to be adjudicated at such time that a dispute arose, and the Treaty is widely viewed in the international community as superseding these claims.
Having said that, it has not stopped countries from taking rather extraordinary steps to bolstering their claims. When Captain Jorge Emilio Palma, of the Argentine research station Esperanza, wife became pregnant she was airlifted by the Argentine government to the station in Antarctica so that the first person born in Antarctica, Emilio Palma would be Argentine. This was a naked attempt at bolstering their sovereignty claim to Antarctica.