I believe that there are the rules in-game, but here are the basics; You place cards on a 3x3 grid. If you place a card next to an opponent's card, and the number on your card is higher then the number on your opponent's card, then you get it. Did that make sense? Probably not. Here is a visual representation.
Your card:
Top:5
Left:4 Right:7
Bottom:3
Your opponent's card:
Top:2
Left:6 Right:9
Bottom:3
If you place your card to the left of his (so that the 7 and 6 are touching), you get that card because seven is greater than 6. He can still get it back by placing a higher card somewhere next to it. If you win, you choose one of their cards to keep. If you lose, you lose a card.
Also, if the winner has captured ALL of the cards on the board, they get to keep them all.
Oh, I get it. It's better than the slot machines in the Johto and Hoenn regions which are just based on luck. This is kind of based on luck too, but at least there's also strategy involved.