“Playing a card” is a lot like playing a card in a real, physical card game. You are taking it out of your (virtual) hand and putting it down on the (virtual) table.

You play a card to make it a part of the current scene. If your character is doing something they are uniquely suited for, you might want to play your Character Type card. If your character fouls something up because of their Weakness card, you should play that card. You can play as many cards as you have in each scene, and you get them back when the scene is over.