1.2.17 The Lord personally cleanses the desire for material enjoyment......

Krishna Himself cleanses the anarthas

śṛṇvatāḿ sva-kathāḥ kṛṣṇaḥ

puṇya-śravaṇa-kīrtanaḥ

hṛdy antaḥ stho hy abhadrāṇi

vidhunoti suhṛt satām

SB 1.2.17: Śrī Kṛṣṇa, the Personality of Godhead, who is the Paramātmā [Supersoul] in everyone's heart and the benefactor of the truthful devotee, cleanses desire for material enjoyment from the heart of the devotee who has developed the urge to hear His messages, which are in themselves virtuous when properly heard and chanted.

The process of hearing itself develops piety. In SB 1.2.16 we read that after serving devotees we get a desire to hear about Krishna. This verse explains what happens next.

Srila Prabhupada mentions in the purport:
But if one is engaged in hearing the glories of the Lord, gradually he realizes his real position. By the grace of God such a devotee gets sufficient strength to defend himself from the state of disturbances, and gradually all disturbing elements are eliminated from his mind.

Specific Anarthas cause unsteady devotional service. Those are listed in the second shower of the Madhurya Kadambini and are also detailed here:

Sravanam-From the Ear to the(cleansing of the) Heart

1. utsaha mayi - enthusiasm indicating pride thinkin that they unlike others have already mastered everything
2. ghana-tarala: moving unsteadily between diligence and negligence
3. vyudh- vikalpa: flickering determination
4. visaya - sangara: an ongoing struggle with sense pleasure
5. niyamaksama : inability to improve devotional practices
6. taranga rangini: delighting in material facilities that bhakti produces

But Krishna is so kind that if a devotee takes to the process of hearing and develops a strong urge to hear His message, He himself cleanses the desire for material enjoyment.

This gives me great hope as even after some years in devotional service I have not been able to overcome any of my anarthas. When I read this verse I feel very hopeful that if I just hear sincerely Krishna will remove all the abhadrani's from my heart and help me fully express my dormant love towards Him through pure devotional service.

HG RadhaKrishna Prabhu mentioned in class that the first quality of a devotee is the desire to hear about Krishna.When eagerness to hear about Krishna is very strong remembrance is very strong. Then the false ego can be cut to pieces- the material mind can be dissolved

1.2.16 How to get affinity for hearing the Lord's message?

śuśrūṣoḥ śraddadhānasya

vāsudeva-kathā-ruciḥ

syān mahat-sevayā viprāḥ

puṇya-tīrtha-niṣevaṇāt

SB 1.2.16: O twice-born sages, by serving those devotees who are completely freed from all vice, great service is done. By such service, one gains affinity for hearing the messages of Vāsudeva.

Some important points from Srila Prabhupada's purport

1.  The servants of God come to propagate God consciousness, and intelligent people should cooperate with them in every respect.
2. By serving the servant of God, one can please God more than by directly serving the Lord. The Lord is more pleased when He sees that His servants are properly respected because such servants risk everything for the service of the Lord and so are very dear to the Lord.
3. By serving the servants of the Lord, one gradually gets the quality of such servants, and thus one becomes qualified to hear the glories of God. The eagerness to hear about God is the first qualification of a devotee eligible for entering the kingdom of God. 

HG Radha Krishna Prabhu also mentioned in his class that we not only get affinity to hear the messages of Vasudeva but we also get the strength to act on what we have heard.

From Sivarama Swami's book "Suddha Bhakti Cintamani" Chapter entitled Eligibility for Devotion One", Pg 85:

"Rendering service to liberated devotees gives neopyhtes the strength to resist their lower conditioned nature and to stick to the higher, pure ideal of pleasing the Lord. Devotees who neglcet to keep the company with superior Vaishnavas , however, are unable to control their lower nature, and their impure faith leads them away from the path of pure devotion to the path of semblance of devotion. In this regard Caitanya Mahaprabhu quotes the following verse to Sanatana Goswami:

SB 5.12.12: My dear King Rahūgaṇa, unless one has the opportunity to smear his entire body with the dust of the lotus feet of great devotees, one cannot realize the Absolute Truth. One cannot realize the Absolute Truth simply by observing celibacy [brahmacarya], strictly following the rules and regulations of householder life, leaving home as a vānaprastha, accepting sannyāsa, or undergoing severe penances in winter by keeping oneself submerged in water or surrounding oneself in summer by fire and the scorching heat of the sun. There are many other processes to understand the Absolute Truth, but the Absolute Truth is only revealed to one who has attained the mercy of a great devotee.

SB 1.2.13 Hear submissively to make advancement

Important points from the purport to the verse:

tac chraddadhānā munayo jñāna-vairāgya-yuktayā

paśyanty ātmani cātmānaḿ bhaktyā śruta-gṛhītayā

The seriously inquisitive student or sage, well equipped with knowledge and detachment, realizes that Absolute Truth by rendering devotional service in terms of what he has heard from the Vedānta-śruti.

Points from Srila Prabhupada's purport

Book Bhagavat and person Bhagavat

1. A third class devotee is one who has knowledge - engages in so called devotional service is not detached from material association
2.  Therefore he has to receive instructions from authoritative sources of Bhagavata - the personality of a devotee and the Book Bhagavatam
3. A sincere devotee must, therefore, be prepared to hear the Vedic literature like the Upaniṣads, Vedānta and other literatures left by the previous authorities or Gosvāmīs, for the benefit of his progress. Without hearing such literatures, one cannot make actual progress. And without hearing and following the instructions, the show of devotional service becomes worthless and therefore a sort of disturbance in the path of devotional service. Unless, therefore, devotional service is established on the principles of śruti, smṛti, purāṇa or pañcarātra authorities, the make-show of devotional service should at once be rejected. An unauthorized devotee should never be recognized as a pure devotee. By assimilation of such messages from the Vedic literatures, one can see the all-pervading localized aspect of the Personality of Godhead within his own self constantly. This is called samādhi.