What is the Story Of Shree Yagnavalkya Rishi giving back his Knowladge to his Guru Shree Vaisampayana (वैशंपायन).
What is the story of Shree Yagnavalkya Rishi and Godess Saraswati?
The story of Yagnavalkya giving back his knowledge is described in this chapter of the Vishnu Purana. As I discuss in this answer, after Vyasa compiled the mantras of the Vedas into the four books that we call the Vedic Samhitas, he taught the Vedas to various disciples, who each started their own Shakha or Vedic school. In particular, he taught the Yajur Veda to his disciple Vaishampayana, who is most famous for narrating the Mahabharata to Arjuna's great-grandson Janamejaya.
Now Vaishampayana once accidentally killed his own nephew, so he asked his disciples to repent for the sin of Brahmahatya on his behalf. Most of the disciples agreed, but his favorite disciple, Yagnavalkya, refused because he did not want to participate in a ritual with the other disciples. Vaishampayana was furious and demanded that Yagnavalkya give back everything Vaishampayana had taught him, and so vomited out the Yajur Veda. Vaishampayana' other shishyas then turned into Tittiris or partridges and ate up the Yajur Veda, which is why one of the Shakhas of the Krishna Yajur Veda goes by the name Taittiriya:
Of the tree of the Yajur-veda there are twenty-seven branches, which Vaiśampáyana, the pupil of Vyása, compiled, and taught to as many disciples. Amongst these, Yájnawalkya, the son of Brahmaráta, was distinguished for piety and obedience to his preceptor. It had been formerly agreed by the Munis, that any one of them who, at a certain time, did not join an assembly held on mount Meru should incur the guilt of killing a Brahman, within a period of seven nights. Vaiśampáyana alone failed to keep the appointment, and consequently killed, by an accidental kick with his foot, the child of his sister. He then addressed his scholars, and desired them to perform the penance expiatory of Brahmanicide on his behalf. Without any hesitation Yájnawalkya refused, and said, "How shall I engage in penance with these miserable and inefficient Brahmans?" On which his Guru, being incensed, commanded him to relinquish all that he had learnt from him. "You speak contemptuously," he observed, "of these young Brahmans, but of what use is a disciple who disobeys my commands?" "I spoke," replied Yájnawalkya, "in perfect faith; but as to what I have read from you, I have had enough: it is no more than this--" (acting as if he would eject it from his stomach); when he brought up the texts of the Yajush in substance stained with blood. He then departed. The other scholars of Vaiśampáyana, transforming themselves to partridges (Tittiri), picked up the texts which he had disgorged, and which from that circumstance were called Taittiríya; and the disciples were called the Charaka professors of the Yajush, from Charańa, 'going through' or 'performing' the expiatory rites enjoined by their master.
Now Yagnavalkya, having lost the knowledge of the Yajur Veda, engaged in Tapasya to Surya the sun god to regain his lost knowledge. Surya appeared before him in the form of a horse, and retaught him the Yajur Veda, but in a different form: while the Krishna Yajur Veda that Vaishampayana had taught him had the Samhita and Brahmana portions all mixed up (as I discuss here), the Shukla Yajur Veda that Surya taught him had the Samhita clearly separated. In any case, because Surya had taught it to Yajnavalkya in the form of a Vaja or horse, one of the Shakhas of the Shukla Yajur Veda is called Vajasaneyaka:
Yájnawalkya, who was perfect in ascetic practices, addressed himself strenuously to the sun, being anxious to recover possession of the texts of the Yajush. "Glory to the sun," he exclaimed, "the gate of liberation, the fountain of bright radiance, the triple source of splendour, as the Rig, the Yajur, and the Sáma Vedas. Glory to him, who, as fire and the moon, is one with the cause of the universe: to the sun, that is charged with radiant heat, and with the Sushumna ray (by which the moon is fed with light): to him who is one with the notion of time, and all its divisions of hours, minutes, and seconds: to him who is to be meditated upon as the visible form of Vishńu, as the impersonation of the mystic Om: to him who nourishes the troops of the gods, having filled the moon with his rays; who feeds the Pitris with nectar and ambrosia, and who nourishes mankind with rain; who pours down or absorbs the waters in the time of the rains, of cold, and of heat. Glory be to Brahmá, the sun, in the form of the three seasons: he who alone is the dispeller of the darkness of this earth, of which he is the sovereign lord: to the god who is clad in the raiment of purity be adoration. Glory to the sun, until whose rising man is incapable of devout acts, and water does not purify, and touched by whose rays the world is fitted for religious rites: to him who is the centre and source of purification. Glory to Savitrí, to Súrya, to Bháskara, to Vivaswat, to Áditya, to the first-born of gods or demons. I adore the eye of the universe, borne in a golden car, whose banners scatter ambrosia."
Thus eulogized by Yájnawalkya, the sun, in the form of a horse, appeared to him, and said, "Demand what you desire." To which the sage, having prostrated himself before the lord of day, replied, "Give me a knowledge of those texts of the Yajush with which even my preceptor is unacquainted." Accordingly the sun imparted to him the texts of the Yajush called Ayátayáma (unstudied), which were unknown to Vaiśampáyana: and because these were revealed by the sun in the form of a horse, the Brahmans who study this portion of the Yajush are called Vájis (horses). Fifteen branches of this school sprang from Kańwa and other pupils of Yájnawalkya.
Concerning the story of Yagnavalkya and Saraswati, that has to do with an additional element to the above story described in this chapter of the Devi Bhagavatam. In this account, when Surya appears before Yagnavalkya, he just tells Yagnavalkya to pray to Saraswati, who actually fulfills Yagnavalkya's boon:
O Nârada! I now describe the Stotra (hymn) of Sarasvatî Devî, yielding, all desires that Yâjñavalkya, the best of the Risis recited in days of yore to Her. The Muni Yajñavalkya forgot all the Vedas out of the curse of Guru and with a very sad heart went to the Sun, the great merit-giving place. There he practised austerities for a time when the Lolâkhya Sun became visible to him, when, being overpowered by great sorrow, he began to cry repeatedly; and then he sang hymns to him. Then Bhagavân Sûrya Deva became pleased and taught him all the Vedas with their Amgas (limbs) and said :-- “O Child! Now sing hymns to Sarasvatî Devî that you get back your memory.” Thus saying, the Sun disappeared. The Muni Yâjñavalkya finished his bath and with his heart full of devotion began to sing hymns to the Vâg Devî, the Goddess of Speech.... Thus saying, the Maharsi Yâjñavalkya, who had observed fasting, bowed down to the Devî Sarasvatî with great devotion and began to cry frequently. Then the Mahâmâyâ Sarasvatî, of the nature of Light could not hide Herself away. She became visible to him and said “O Child! You be good Kavîndra (Indra of the poets).”
But the Devi Bhagavatam is widely considered not to be either an authentic Mahapurana or Upapurana, so take this with a grain of salt.