What does adding a star to a dragon actually do?

Adding a star does far more than inflate a number. Concretely, each star:

• Raises every stat on the dragon (Strength, Instinct, Intelligence, Initiative)
• Unlocks a Habit slot (roughly one Habit every 2 ranks)
• Increases the army size the dragon can lead
• Changes its size and look

The part people miss: stars also unlock stronger versions of abilities. A confirmed example on Crimson: its stun goes from 20% to 40% on the first turn at 10★, and adds a −12% Instinct/Initiative debuff. Plenty of big effects sleep at 6★ or 10★ until you get there. Permanent, so it's a safe investment.