In his Bhakti-sandarbha, Sri Jiva Goswami Prabhupada writes the following about diksa: divyam jnanam yato dadyat kuryat papasya sanksayam, “(Diksa) involves giving of transcendental knowledge by which sins are destroyed.”
The idea is that the seed of transcendental knowledge that includes knowledge of one’s eternal relationship with Krsna is within the mantra imparted by the guru to the disciple at the time of diksa. This seed of transcendental knowledge has the potential to destroy ignorance (avidya), which is the cause of all distress (klesa) manifesting as one’s karma.
Thus one who is properly initiated is freed from karma to the extent that he or she takes advantage of diska, causing the seed of transcendental knowledge to grow and bear fruit. This takes time. Initiation is not completed in a day. It is completed when one’s knowledge of his relationship with Krsna is realized and one’s sambandha jnana (a devotional conceptual orientation to life, literally, knowledge of relationship) is thus complete. Diksa is a function of sambhanda jnana. When one’s sambandha jnana is complete, one can graduate from sadhana bhakti. Bhava bhakti thus involves devotional engagement with the realization of one’s relationship with Krsna, which is cultivated in emotional ecstacy up to the stage of prema.
So, the initiated devotee is free from karma to the extent that he applies himself to that which is given to him at the time of initiation: Krsna nama and Krsna mantra. Furthermore, whatever karmic influence remains in his life is an edited form of the karma that he is due, edited by Krsna to bring his devotee closer to him.