In Paper Mario: Color Splash, Mario must collect 6 Paint Stars, formally called Big Paint Stars. They are the central collectibles driving the game's story and progression across Prism Island.
At the beginning of the adventure, the Prisma Fountain is dry and the island has been drained of its color. Huey, the fountain's guardian, explains that the six Big Paint Stars normally supply Prism Island with its paint and color. Mario therefore travels through six major areas, recovering one star from each area while helping local Toads and confronting enemies, including the Koopalings.
The stars are more than ordinary collectibles. As Mario retrieves them, they gradually reveal what happened to Port Prisma. Bowser had tried to dye his shell rainbow-colored using the fountain, accidentally mixing its colors into toxic black paint. The black paint possessed him and caused the island-wide crisis.
Six is sometimes confused with the game's many Mini Paint Stars, which unlock individual stages and can appear multiple times within a level. The question refers to the six major Big Paint Stars, not the smaller stage-ending collectibles.