Being broken basically means that you are either mentally or physically damaged in some way, either from illness or another another cause. You may be brokenhearted, broken in spirit, or even broken in a medical sense. Being broken means that you are hurting in some way but things will get better with time in most cases. Facing the hardships of life head on and overcoming them is what makes us stronger.
I'm selfish, impatient, and a little insecure. I make mistakes, I am out of control and at times hard to handle. But if you can't handle me at my worst, then you sure as hell don't deserve me at my best.