So Admiral Byrd says "an area as big as the United States that has never been seen by human being and that's beyond the Pole on the other side of the South Pole from Little America"
Below is a map showing where their base called "Little America" was located on the Ross ice shelf. The edge of the continent there is close to the South Pole whereas the bulk of Antarctica extends the other side of the South Pole. That is the unexplored territory Admiral Byrd is referring to, because apart from where their base was established which was relatively close to the South Pole, allowing them to fly to the South Pole and back, the rest of the operation only surveyed the coastline. They never explored deeper into Antarctica.
CF won't let me link to the image inline so you will have to click on the link to see the map.
Admiral Byrd also refers to Antarctica in the interview as the "bottom of the world", so he clearly understands the Earth to be a globe.