App Store purchase: sign in with the same Apple Account you bought with, install Aura, and tap Restore Purchases on the paywall. Apple returns your subscription.
Website purchase: sign in to Aura with the same email you bought with, and your subscription comes with the account.
If restore doesn't find anything, the most common reason is being signed in to a different Apple Account than the one that made the purchase. Check iOS Settings, tap your name, and confirm which account you're on.
Your habits, streaks, and coins return with your account. Wall of Wins photos stay on the old phone because they are stored locally.