Basically, there are four numbers on each card. If the number of a card you put down is larger than an adjacent number on an adjacent card, the card with the lower value gets flipped and becomes yours.
Okay, that sounds slightly complicated, so here's a diagram:
Let's say player 1 puts down this card, with a birdie on it:
Then player 2 puts down this card to the right, with the freaky gray guy:
The 8 on the left of the Gray Guy card is larger than the 4 on the right of the Bird card. Therefore, the Bird card is flipped over and now belongs to player 2.
Repeat until all the spaces on the 3 by 3 grid are filled, the player with the most cards belonging to him/her at the end is the winner.
That's the basic rule, there are other, more complicated ones, but you won't need to know them immediately.