Meditation cultivates inner peace that we bring into the world. Peacefulness is energy. Others can feel this energy. You really cannot change other people, but you can change the energy that you bring to others. Over time, this energy of peace is going to impact your relationships.
The first relationship that meditation is going to change is the relationship you have with yourself. While sitting in meditation with yourself, you are getting to know yourself in a deeper way. While sitting with yourself, your emotions, your reactions to emotions, you discover your inner landscape. You learn what your triggers are. Sitting with your triggers, you uncover the root causes of your pain to understand, forgive and heal.
Once you understand yourself deeply, you begin to see yourself in others. Judgement is replaced with love.