As the name suggests, Activity Points are required by some activities or facilities in the Monastery when you’re in Exploration mode. This limits what you can or cannot do during your day off. As such, it’s really important to carefully consider what activity you’ll do to avoid wasting your weekend. You can see your activity point by just standing still or when you're about to select an action that requires an activity point.
You’ll start off with one Activity Point but this can be further increased as your Professor Level gets higher. When your activity points run out, you can still explore the monastery as much as you want but for the day to move on, you have to press L to end your free time to proceed to next week.
Take note that there are some activities that doesn’t consume Activity Points such as Fishing, checking the Advice Box, Giving Gifts to NPCs, etc. You’ll need to maximize your time every chance you get, especially if you’re planning to Recruit as many characters as you can before the time skip plot.