How does triple level cell FLASH memory achieve 3 bits per cell?
According to the source you linked,


  Samsung announced a type of NAND flash that stores three bits of information per cell, with eight total voltage states. This is commonly referred to as Triple Level Cell (TLC)...  [emphasis added]


So it stores 3 bits by using 8 voltage levels, just as you'd expect; not 3 levels as your question text claims.

As for why they called it "triple-level" and not "octa-level", probably the marketing guys just thought "triple-level" sounded better.