Who is the incarnation that the  Vedic scriptures do not openly talk about?
Sri Caitanya Mahaprabhu-Is He an Incarnation?
In this Kali-yuga the incarnation is Lord Caitanya Mahaprabhu. It is stated in the Shastra, krishna-varnam, Krishna varnayati: “He is simply chanting ‘Krishna,’ although He is Krishna Himself.” Krishnam-varnam tvisakrishnam. Therefore He has hidden Himself. His bodily color is more Krishna, but golden. Krishna has got four colors: Suklo raktas tatha pita idanim krishnatam gatah. So Krishna has another color: pita. Pita-varna-gaurah. That is Lord Caitanya. Therefore tvisa akrishna: “Bodily complexion, not Krishna.” Sangopangastra-parsadam: Followed by many devotees. – Lecture on Srimad-Bhagavatam 1.16.3, Los Angeles, December 31, 1973
The loving affairs of Sri Radha and Krishna are transcendental manifestations of the Lord’s internal pleasure-giving potency. Although Radha and Krishna are one in Their identity, previously They separated Themselves. Now these two transcendental identities have again united, in the form of Sri Krishna Caitanya.- Cc. adi 1.5
Why Hidden?
Caitanya Mahaprabhu is the Supreme Personality of Godhead, but He’s channa. Channa means “covered.” Lord Caitanya is a covered avatara, or incarnation, of the Supreme Lord because He appeared as a devotee. In the Kali-yuga He does not appear directly like other incarnations, such as Narsimhadeva, Vamanadeva, or Lord Ramachandra. He appears as a devotee. Why? Because He is the most magnanimous avatara. People are so foolish that they could not understand Krishna. Sri Caitanya Mahaprabhu appeared to give mercy to the fallen souls, who are so foolish that they cannot take to Krishna consciousness. He is personally teaching how to take to Krishna consciousness. And that process is this kirtana, chanting Krishna’s holy names. – Lecture in Mayapur on the appearance day of Lord Sri Caitanya Mahaprabhu: March 16, 1976
You appear in various incarnations as a human being, an animal, a great saint, a demigod, a fish, or a tortoise, thus maintaining the entire creation in different planetary systems and killing the demoniac principles. According to the age, O my Lord, You protect the principles of religion. In the Age of Kali, however, You do not assert Yourself as the Supreme Personality of Godhead, and therefore You are known as Triyuga, or the Lord who appears in three yugas. – Srimad-Bhagavatam 7.9.38
The Lord tried to conceal His identity as the Supreme Personality of Godhead by representing Himself as a devotee, but His pure devotees could recognize Him by His special features. The Vedas and Puranas foretell the appearance of Lord Caitanya, but still He is sometimes called, significantly, the concealed descent of the Supreme Personality of Godhead. – Introduction to Cc. adi Chapter 3
External Reasons for Lord Caitanya’s Appearance
According to the Vedic literature, the foremost occupational duty for humanity in this Age of Kali is nama-sankirtana, or congregational chanting of the holy name of the Lord. The incarnation for this age especially preaches this process, but only Krishna Himself can explain the confidential loving service performed in the four principal varieties of loving affairs between the Supreme Lord and His devotees. Lord Krishna therefore personally appeared, with His plenary portions, as Lord Caitanya. As stated in this chapter, it was for this purpose that Lord Krishna appeared personally in Navadvipa in the form of Sri Krishna Caitanya Mahaprabhu.
Advaita acharya felt sorry for the condition of the world because even after Lord Krishna’s appearance, no one had interest in devotional service to Krishna. This forgetfulness was so overwhelming that Advaita Prabhu was convinced that no one but Lord Krishna Himself could enlighten people about devotional service to the Supreme Lord. Therefore Advaita requested Lord Krishna to appear as Lord Caitanya. Offering tulasi leaves and Ganges water, He cried for the Lord’s appearance. The Lord, being satisfied by His pure devotees, descends to satisfy them. As such, being pleased by Advaita acharya, Lord Caitanya appeared. – Introduction to Cc. adi Chapter 3
Confidential Reasons for Lord Caitanya’s Appearance
The first purpose was to relish the position of Srimati Radharani, who is the prime reciprocator of transcendental love of Sri Krishna. Lord Krishna is the reservoir of transcendental loving transactions with Srimati Radharani. The subject of those loving transactions is the Lord Himself, and Radharani is the object. Thus the subject, the Lord, wanted to relish the loving mellow in the position of the object, Radharani.
The second reason for His appearance was to understand the transcendental mellow of Himself. Lord Krishna is all sweetness. Radharani’s attraction for Krishna is sublime, and to experience that attraction and understand the transcendental sweetness of Himself, He accepted the mentality of Radharani.
The third reason that Lord Caitanya appeared was to enjoy the bliss tasted by Radharani. The Lord thought that undoubtedly Radharani enjoyed His company and He enjoyed the company of Radharani, but the exchange of transcendental mellow between the spiritual couple was more pleasing to Srimati Radharani than to Sri Krishna. . . . Krishna is the transcendental male, and Radharani is the transcendental female. Therefore, to know the transcendental pleasure of loving Krishna, Lord Krishna Himself appeared as Lord Caitanya, accepting the emotions and bodily luster of Srimati Radharani. – Introduction to Cc. adi Chapter 4
Activities of Lord Caitanya
After accepting the renounced order of life, Caitanya Mahaprabhu converted many karma-nisthas who were addicted to fruitive activities, many great logicians like Sarvabhauma Bhattacharya, nindakas (blasphemers) like Prakasananda Sarasvati, pasandis (nondevotees) like Jagai and Madhai, and adhama paduyas (degraded students) like Mukunda and his friends. All of them gradually became devotees of the Lord, even the Pathans (Muslims). – Cc. adi 7.39, purport
Imparting Transcendental Knowledge
For ten days Sri Caitanya Mahaprabhu stayed at Prayaga and instructed Rupa Goswami, empowering him with the necessary potency. -Cc. adi 19.135
Inaugurating the Sankirtana Movement
Sri Caitanya Mahaprabhu wanted to preach the sankirtana movement of love of Krishna throughout the entire world, and therefore during His presence He inspired the sankirtana movement. 
– Cc. adi 7.171, purport
Delivering the most fallen
Lord Sri Caitanya Mahaprabhu has freely given this love of Krishna everywhere and anywhere, even to the most fallen, such as Jagai and Madhai. What then to speak of those who are already pious and elevated? – Cc. adi 8.20
Lord Caitanya’s Special Forms
Sarvabhauma Bhattacharya was so fortunate as to see the six-armed form of Lord Caitanya known as sadbhuja. A sadbhuja Deity is still situated at one end of the Jagannatha temple.
Lord Caitanya’s Preaching Mission
Sri Caitanya Mahaprabhu wanted the cult of His sankirtana movement to spread all over the world. Now that the International Society for Krishna Consciousness has taken up this task of preaching the cult of Lord Caitanya, its members should not only construct temples in every town and village of the globe but also distribute the books that have already been written and further increase the number of books. – Cc. adi 7.164, purport
Krishna-varnam tvisakrishnam sangopangastra-parsadam/ yajnaih sankirtana-prayair yajanti hi su-medhasah: “In the age of Kali, intelligent persons perform congregational chanting to worship the incarnation of Godhead who constantly sings the names of Krishna. Although His complexion is not blackish, He is Krishna Himself. He is accompanied by His associates, servants, weapons and confidential companions.” – Srimad-Bhagavatam 11.5.32