Adventure mode is unlocked by getting a D rank or higher in (IIRC) the Dungeon level in Score Attack. There are six courses with a boss at the end of each and boss battles are explained in the second tutorial:
This achievement requires defeating the boss in area 1-11. You need 10 stars from the other areas in order to unlock 1-11, meaning you can get one star in all 10 areas or multiple stars across a few (you eventually need 3 stars in all areas to complete Adventure Mode).
The gist of the boss level is to eat pellets until the fruit appears (at which point two 1-ups will appear, as well). This angers the ghosts, which you need to now avoid while chasing the fruit to then move to the next stage. Complete stages until you eat enough fruit, after which you'll play a stage with a power pellet. Eat enough pellets to make the power pellet appear, then eat the power pellet to defeat the boss.
Simply surviving to the end and eating the power pellet (a two star run) is sufficient for this achievement. A six star run requires collecting the 1-ups that appear on each stage along the way and having enough lives left at the end: