We have 108 beads because there are 108 Upanisads. By this chanting on the 108 beads we have surpassed the study of 108 Upanisads by realizing the conclusion of the 108, namely to be absorbed in the devotional service of Lord Sri Krishna.
Also there are 108 gopis in Vrindavan who are especially dear to Krishna. So by chanting on these 108 beads we are serving those 108 chief gopis. Serving Krishna's devotees is essential because becoming a devotee of Krishna means to become a devotee of the devotees of Krishna.