Star ranking is the secondary method of improving your cards, and unlocks stat improvements and new skills.
Increasing star ranking consumes Secrets of Stars scrolls, or more simply scrolls, which can be acquired through witch activities, hard quests (on first completion), and examination resource quests.
A card’s star ranking affects their level cap (in combination with your player ranking), unlocks new skills, and increases stats.
For SSR and UR cards, star rank 1 is incredibly important as it unlocks their enhanced attack, which is vital for their use.