What exactly does it mean that Japan is the "land of the rising sun"?
Nippon means "sun origin" or "land of the rising sun". From the direction and point of view of the older, larger civilisation of the continent to the West - China - the island of Japan were in the general direction of where the sun rises, with no known lands further East beyond them.

China was a powerful cultural influence for Japan. Nihon/Nippon was a nicer, more poetic name than the Chinese name of "Wa" - which was Chinese for "Dwarf"; Nihon/Nippon ("sun origin") was used, possibly intentionally, to supplant it within Japan. 

I don't know if Japanese or Chinese historically understood that Earth was a globe but suspect it was largely irrelevant to that naming.