How does mindfulness-based cognitive therapy help addiction?

Self soothing. Mindfulness allows you to correct inaccurate beliefs and feelings triggered by events before you self medicate the negative feelings that arise. Mindfulness and meditation are effective methods to self soothe and thereby avoid relapse.
Adjuctive. Mindfulness practices can be very useful to develop coping strategies. Think of these as helping to deal with negative thoughts and emotions in a healthy way rather than turning to the addiction.