The occurrences of Hayagrīva in the Mahābhārata (henceforth MBh) have been neatly summarised in van Gulik 1935, pp. 10–15 and in Nayar 1994, chapter 3. Van Gulik notes that in different portions of the Mahābhārata we find Hayagrīva connected with the recitation of the Vedas and that in MBh 12.335.43–69 Viṣṇu horse-headed brings back the Vedas and kills their thieves, the two asuras Madhu and Kaiṭabha, who had stolen them from Brahmā. The following is an excerpts of the main action (my tentative translation):

Having entered the mythical stream, [Viṣṇu-Hayagrīva] performed the supreme Yoga |

Performing the sound according to the rules of phonetics, he pronounced the Oṃ || 12.353.50 ||

The sound was resonant and went in each direction and was charming |

It was in the whole earth and had all good qualities || 12.353.51 ||

Then, the two asuras, made up an agreement regarding the Vedas (presumably: regarding when to come back and pick them up) |

and having threw them on the bank of the mythical stream, they run whence the sound came from || 12.353.52 ||

At that point, the king god carrying a horse head, |

Hari, grasped all the Vedas which had arrived to the bank of the mythical stream || 12.353.53 ||

He gave them back to Brahmā and went then back to his own nature |

[…] Then, the two [demons] sons of Danu, Madhu and Kaiṭabha, who did not see anything [as the source of the charming sound they had head before] |

went back quickly to the place [where they had left the Vedas] and they looked || 12.353.55 ||

Where the Vedas had been thrown, the place was empty! |

[…] Then there was a fight between them and Nārāyaṇa |

The two Madhu and Kaiṭabha, whose bodies where filled with rajas and tamas, |

were killed by the [now become] ‘Killer of Madhu’ (Madhusūdana, a name of Viṣṇu), who thereby pleased Brahmā || 12.335.64 ||

rasāṁ punaḥ praviṣṭaś ca yogaṁ paramam āsthitaḥ |
śaikṣaṁ svaraṁ samāsthāya om iti prāsr̥jat svaram || 12.353.50 ||
sa svaraḥ sānunādī ca sarvagaḥ snigdha eva ca |
babhūvāntarmahībhūtaḥ sarvabhūtaguṇoditaḥ || 12.353.51 ||
tatas tāv asurau kr̥tvā vedān samayabandhanān |
rasātale vinikṣipya yataḥ śabdas tato drutau || 12.353.52 ||
etasminn antare rājan devo hayaśirodharaḥ |
jagrāha vedān akhilān rasātalagatān hariḥ |
prādāc ca brahmaṇe bhūyas tataḥ svāṁ prakr̥tiṁ gataḥ || 12.353.53 ||
[…]
atha kiṁ cid apaśyantau dānavau madhukaiṭabhau |
punar ājagmatus tatra vegitau paśyatāṁ ca tau |
yatra vedā vinikṣiptās tat sthānaṁ śūnyam eva ca || 12.353.55 ||
[…]
atha yuddhaṁ samabhavat tayor nārāyaṇasya ca || || 12.353.63 ||
rajastamoviṣṭatanū tāv ubhau madhukaiṭabhau |
brahmaṇopacitiṁ kurvañ jaghāna madhusūdanaḥ || 12.353.64 ||

The connection with the Veda, perhaps both with their oral and written form (although it is possible that what is rescued is still an oral version of the Vedas), is here very evident. It is also interesting that this version of the rescue of the Vedas is the only one which will be referred to in Pāñcarātra and in Viśiṣṭādvaita Vedānta texts. I will also come back (in a future post) to the motif of the ocean, which is sometimes connected with Hayagrīva (although the word rasā might also mean ‘lower regions, hell’, its connection with tala `bank’, as well as the evidence derived from parallel texts, seem to suggest the meaning ‘stream’). However, the rationale for the fact that Viṣṇu assumed exactly a horse head is altogether absent (unlike in the case of his Matsya or Varāha-avatāras, where the transformation had to do with the task to be accomplished).

Another mention of Hayagrīva in the MBh has it figure as the name of a demon slaughtered by Viṣṇu:

The two Madhu and Kaiṭabha have been slain by [Viṣṇu], who lies on the ocean |

Having reached a different birth, Hayagrīva has also been slain in the same way || 5.128.49 ||

Instead, the following one is a summary of the Hayagrīva story in one of its Purāṇic forms:

A horse-headed Asura called Hayagriva once invoked Brahma and sought from him [\dots] a boon by which he could be defeated by none other than another being who also had a horse’s head, also called Hayagriva. Such a creature did not exist […] The Devas did not know what to do. […] When they went to Vishnu, they found him taking a nap, resting his chin on his bow. Taking the form of termites, the Devas ate into the bowstring so that the bow shaft snapped with such force that it severed Vishnu’s neck. To save the headless Vishnu, the Devas sacrificed a horse and placed its head on his neck. Vishnu thus transformed into a horse-headed being. […] Vishnu challenged Hayagrīva to a duel, smote him with his mace and restored the Veda. […] Brahma then restored Vishnu’s head. (Skanda Purāṇa). (Pattanaik 2006, s.v)

There are various versions of this story (other versions have, e.g., Viṣṇu loose his head because of a curse and involve no good finality for it, see Nayar 1994, chapter 3) and in any case the story looks somehow strange, since:

• it looks like an ad hoc explanation for Viṣṇu’s horse head
• it looks like the conflation of three different stories, i.e., the slaughter of the demon Hayagrīva, the slaughter of Madhu and Kaiṭabha, who had stolen the Vedas, and the slaughter of the demon Hiraṇyakaśipu. As for the latter, according to the Bhāgavata Purāṇa, after years of ascesis, the demon Hiraṇyakaśipu had obtained from Brahmā a boon of his choice and asked for immortality, but Brahmā refused. Therefore, Hiraṇyakaśipu asked to be killed neither by a human being nor by an animal, nor by a demon, nor by a God. He is at last killed by Viṣṇu in the form of Narasiṃha, who is neither a human being, nor an animal, nor a God. The request by Hayagrīva seems very similar.

It may be objected that once one does not accept the Purāṇic versions of the story, it is difficult to make sense of Viṣṇu’s horse head. In fact, this might be due to either an ancient (Vedic or perhaps Indoeuropean) attribute of a deity, linking it to the horse because of the latter’s importance in the Vedic mythology or the inclusion of a pre-existing deity in the Smārta pantheon through the device of turning it into an avatāra of Viṣṇu.

Thus, in my opinion Hayagrīva is a (perhaps Vedic) deity, perhaps assimilated to Viṣṇu or always identical with him, and the horse head is linked to the importance of the horse in the Vedic culture. The same importance has led to the invention of several demons with horse attributes, until someone conflated the two stories into one, with added details from other demons’ slaughters (Madhu and Kaiṭabha and Hiraṇyakaśipu).

On Hayagrīva see also this post (about the Hayaśīrśa Saṃhitā) and this one (about Viśiṣṭādvaita Vedānta texts on him).