Leveling Up is a process in the Paper Mario series in which Mario's level goes up by one.
In Paper Mario and Paper Mario: The Thousand-Year Door, leveling up only happens when Mario earns 100 star points. It also allows Mario to increase his HP or FP by 5, or BP points by 3. Also, if Mario's HP, FP, and star power is low, is also heals that too.
SInce star points don't appear in Super Paper Mario, the process works differently. Since the game is based off of both the Paper Mario series and the Mario Bros. series, leveling up is based on the amount of points that Mario has. In the main menu, it shows how many points Mario needs to earn in order to level up. Once he reaches the amount of points, his level goes up by one and his HP increases by 5.