Question
Why did Lord Chaitanya appear on earth?
Answer
Harinama Sankirtana (chanting of the Lord's names) is the prescribed spiritual duty for the people of Kali Yuga:
kṛte yad dhyāyato viṣṇuṁ tretāyāṁ yajato makhaiḥ
dvāpare paricaryāyāṁ kalau tad hari-kīrtanāt [SB - 12.3.52]Meaning
Whatever result was obtained in Satya-yuga by meditating on Viṣṇu, in Tretā-yuga by performing sacrifices, and in Dvāpara-yuga by serving the Lord’s lotus feet can be obtained in Kali-yuga simply by chanting the name of Hari.
Although the scriptures defined the prescribed duty, they didn't tech how to do it practically. For this pupose a teacher is required. And who could other than the God himself can teach us how to practice His nama samkirtana? So God from time to time takes birth here for teaching us the practical process:
martyāvatāras tv iha martya-śikṣaṇaṁ
rakṣo-vadhāyaiva na kevalaṁ vibhoḥ [SB 5.19.5]Meaning
But God takes avatars here to teach or educate the people, not only just to kill the demons.
It is for this reason that Lord Chaitanya took birth here. He revived the samkirtana movement, brought into public the Hare Krishna mahamantra, and staying as a sannyasi Himself, He taught others how to practice devotion through the process of chanting the Lord's names and forms. In those mediaval times irreligious activities had taken over and whether or not Lord Chaitanya's divinity is accepted, His activities and pastimes restored the yuga dharma and many people are getting the opportunity of chanting the Lord's names now.
So the Chaitanya Charitamrita states Shir Krishna's desire for taking birth as a devotee:
yuga-dharma pravartāimu nāma-saṅkīrtana
cāri bhāva-bhakti diyā nācāmu bhuvana [CC - 1.3.19]Meaning
I shall personally inaugurate the religion of the age — nāma-saṅkīrtana, the congregational chanting of the holy name. I shall make the world dance in ecstasy, realizing the four mellows (dāsya, sākhya, vāstalya, mādhurya) of loving devotional service.
So this was the primary reason behind Lord Chaitanya's appearance from Gaudiya Vaishnava perspective. You can read the third and fourth chapter of Chaitanya Charitamrita to know more about this.
Regarding Him being an avatara of Vishnu, people have different views. But in Gaudiya Vaishnavism, He is accepted as an incarnation of Vishnu or Shri Krishna [CC - 1.5.6]. Regardless of His divinity, He is revered as a spiritual teacher by whose mercy one crosses this material ocean filled with crocodiles of various opinions:
śrī-caitanya-prabhuṁ vande bālo ’pi yad-anugrahāt
taren nānā-mata-grāha-vyāptaṁ siddhānta-sāgaram [CC - 1.2.1]Meaning
I offer my obeisances to Sri Caitanya Mahāprabhu, by whose mercy even an ignorant child can cross the ocean of theories filled with crocodiles of various opinions.
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