They act as the Tattle Log of enemies, as opposed to Tippi actually using Tattle in battle. They give you info on enemies and other characters in the game. They give Hp, attack, and a short description.
Another thing they do is double the damage done to the enemy you have the card of. It works for how many cards you have. 1 card equals 2x damage, 2 cards equal 4x damage, and so on. It is exponential based on how many cards you have.
Catch cards can be bought or found blank or caught. Caught cards are the ones that already have an enemy or character on it. Blank cards are items you carry and try to use on an enemy to get their catch card. They don't work as good as a Catch Card SP. They almost always work except on bosses and characters.
There are 2 cards shops each on the first floor of Flipside and Flopside. The one in Flipside, Catch a Dream, sells catch cards for 20 coins and caught cards for 10 coins. The caught cards are random and not reveiled until you pay for it. The one in Flopside, Fondest Hopes, sells Catch Card SPs for 100 coins and caught cards at random prices. These cards are shown to you but they are expensive, usually at least 50 to 100 coins.
Character cards are obviously useless in battle. They are for bragging rights and for selling them. You can sell the cards to both shops for the same price. Rarer cards are sold and bought for more. The cards have a rarity system. Non-boss enemies are yellow with one star. Bosses and characters are silver with 2 stars. The rarest of enemies and Pixls are gold with 3 stars.
There are a total of 256 catch cards total. They are a major side quest but there isn't an actual reward for collecting them all.