To get higher ranks, get the required amount of Rescue Points.

Unlockable - How to Unlock

Bronze Rank - Get 50 Rescue Points
Diamond Rank - Get 7500 Rescue Points
Gold Rank - Get 1500 Rescue Points
Platinum Rank - Get 3000 Rescue Points
Rukario Rank - Get 15000 Rescue Points
Silver Rank - Get 500 Rescue Points