Being "in the doghouse" is an English expression for being in trouble with someone. For example: I was in the doghouse when my mom came home and the dishes weren't done! In the example, the writer is in trouble with his/her mom because he/she didn't do a task. So, a "get out of the doghouse free" card might be an offer from the person you are in trouble with to resolve the problem easily. Another expression that is similar to that one is "get out of jail free card," like in the board game Monopoly. Basically, it's just an easy way to get out of trouble.