From the 5th level you can collect keys and enter new courtyard sections of your opponents with even phat loot. Also, some bosses require you to have keys in order to fight them at all.
There are two ways to get keys:
1. Insert the scanner. The scanner will scan everything in the opponent's yard and you can see directly which cat or pile of fish a key is hidden under. The key, once scanned, becomes visible over a cat's head or over a pile of fish.
2. You can also just guess where the key is. If you loot a pile of fish with a key, or take out a cat that's carrying a key, you'll get it.
As simple as that!