Compassion is not feeling pity for someone. It is the understanding that the other person wants the same things we do—to be free of suffering and to find genuine happiness. Our compassion is our wish and intention for them to attain this. It is to regard the other with respect and gratitude for the gift they are giving us.
What gift? They are helping us to open up to another being, to not judge or dismiss, but to selflessly want to help. They are teaching us to not impose conditions, but to give selflessly of ourselves.
The result? We have gotten just a little bit better at perfecting our compassion and have moved just a little bit further along the Path. So actually, we are not helping the other. They are helping us.