The verb 落ち着く means 'to calm down', but neither constituent pertains to any sort of emotion. Was there originally a subject or object (e.g. temper coming down)?
落ちる means "to go down" and 着く means "to arrive" or "to settle", and 落ち着く is a rather simple combination of the two verbs. Just like English "settle" has multiple meanings, 落ち着く can mean "to (psychologically) settle down", "to establish in residence", "to become stable", "to establish a conclusion", etc.