Each time you play through a stage, you will see an increase in the number of days (can be seen in the bottom right corner of the menus).
To unlock this achievement, you must play through stages a total of 365 times.
Please note that simply restarting a stage without finishing it will not count towards days. The stage must be completed (mode and grade does not matter).
Although this will come naturally as you progress through the game, you can always boost it by playing and replaying any short stage (i.e first driver ed course).