In order to have compassion for others, we have to have compassion for ourselves.
In particular, to care about other people who are fearful, angry, jealous, overpowered by addictions of all kinds, arrogant, proud, miserly, selfish, mean —you name it— to have compassion and to care for these people, means not to run from the pain of finding these things in ourselves. In fact, one's whole attitude toward pain can change. Instead of fending it off and hiding from it, one could open one's heart and allow oneself to feel that pain, feel it as something that will soften and purify us and make us far more loving and kind.
The tonglen practice is a method for connecting with suffering —ours and that which is all around us— everywhere we go. It is a method for overcoming fear of suffering and for dissolving the tightness of our heart. Primarily it is a method for awakening the compassion that is inherent in all of us, no matter how cruel or cold we might seem
We begin the practice by taking on the suffering of a person we know to be hurting and who we wish to help. For instance, if you know of a child who is being hurt, you breathe in the wish to take away all the pain and fear of that child. Then, as you breathe out, you send the child happiness, joy or whatever would relieve their pain. This is the core of the practice: breathing in other's pain so they can be well and have more space to relax and open, and breathing out, sending them relaxation or whatever you feel would bring them relief and happiness. However, we often cannot do this practice because we come face to face with our own fear, our own resistance, anger, or whatever our personal pain, our personal stuckness happens to be at that moment.