• The game has four colored suits, but the numbered cards don't have suits, with five copies of each number.
• When you play a card, you declare which color it is, marking that specific number-color pairing on a game board with a token, with you trying to group lots of your tokens together.
As in most trick-taking games, you need to follow the suit led to the trick, but since your cards have no color, you can declare yourself out of a color and play another color, whether the trump suit (red) to claim a trick that would have otherwise gotten away or to play a number already claimed in that color so that you can place a token in a vital location.
This choice has consequences, however, as you cannot play that color for the remainder of the hand. If you end up with a 6 and 8 in hand on the final trick of the round, and the only places you can claim a 6 or 8 on the game board are in that forbidden color, then you cannot play, the trick ends immediately, and you lose points for each trick you claimed.
I've played Cat in the Box seven times — three times on the original design with 3 and 4 players, and four times on a review copy of the Cat in the Box: Deluxe Edition from Bézier Games with 2, 3, and 5 players — and while the game uses Schrödinger's Cat as a reference point for the gameplay, with cards having all colors until you declare them to have a specific one, it could instead have been based on psychology, driven by the notion that your past actions have consequences that will affect you in the future.
In almost every hand, you face a moment or two when you need to decide whether to abandon a color that you could play in order to make what you think will be a better play — and if you do, then a few turns later you're cursing your past self for putting yourself in a difficult situation, taking a trick you don't want because now you won't make your bid exactly, which means you forfeit bonus points based on how you've grouped your tokens. Worse, you can't play at all, and everyone else profits while you burn.
In the video below, I run through the details of gameplay, explain how the two-player game creates a tight box in which to fight, and detail the pluses and minuses of the components in the deluxe edition: