This is a difficult idiom to explain. When you "play the X card" you are using something you have to create an advantage for yourself, usually by creating sympathy.
Played the grieving widow card, played the victim card, played the heart transplant cart... all of these people have suffered in a real way, but the use this idiom indicates that the speaker thinks that the person is using the situation to his or her advantage to a level beyond what is "fair."