THERE ARE TWO VERSIONS OF THIS STORY, THIS IS THE VERSION TOLD IN HÁVAMÁL.
Stanza 104
“I once journeyed to the hall of an old giant, and I’ve since returned. I did not sit silently during my visit, I spoke many words in support of my quest in the halls of the great Suttung.”
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This artwork depicts Odin disguised and talking to Baugi, which isn't actually mentioned in the Hávamál version of the story. It is interesting though, as he slays Baugi's farmhands in secret, and then shows up to help' Baugi, brother of Suttung, and in return would be granted an audience with the giant who holds the mead of poetry.
Stanza 105
“His daughter, the beautiful Gunnloð, gave me a drink, of her fathers precious mead, while I sat like a kind in a golden chair. I would later betray her kindness, with a terrible repayment for her trusting soul, and I would leave her spirit troubled."
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After being denied this audience, Odin would use Baugi's help to sneak into the dim and damp cave where Gunnloð would reside. Shapeshifting into a snake to slither through a small gap, Baugi would close the gap behind him, realising just who he was helping.
Once inside, Odin would proceed to seduce Suttung's daughter, the beautiful giantess Gunnloð. She would become infatuated with him, and he would lay with her for three nights in return for a sip from each of the three vats of mead.
Stanza 106
"surrounded on all sides by the giant's dwelling, I would use Rati's tusk to burrow out from these rocky halls. I risked my life in this escape, with the mad stowed within my body."
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Odin would consume the entire contents of each vat with each sip, proceeding to sneak away during the night. Using a bore called 'Rati's tusk, he would burrow a hole through the side of the mountain. this would be his escape.
Once free, he would shape shift into the form of an Eagle, hastily flying towards the rim of Midgard, and then up into Asgard where the gods awaited with several large vats to store the mead.
Stanza 107
"I made great use of my disguise. None were aware that it was Odin who stood within their halls. No task is too difficult for the wise. Now in the form of Oðerir, I flew to the rim of Midgard, the halls of Asgard in my sight."
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Suttung would see the Eagle flying away. Though he did not know his precious mead had been stolen, he was suspicious none the less.
To avoid being caught, Odin would hastily spit the mead into the vats laid out by the gods.
In such a haste, a few drops fell into midgard. It is said that the abilities of human skalds and artists have since come from these measly drops.
Stanza 108
“I doubt I would have escaped those halls of Jotunheim, if I hadn’t used Gunnloð. She was a pure woman, who rested in my arms and trusted my intentions”
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Reflecting on his ordeal, Odin realises that if not for how he seduced, used and betrayed Gunnloð, he never would have escaped from Jotunheim.
He likely would have been found by the powerful giants, and as a result would have lost his head.
Stanza 109
“The very next day, the giants travelled to the Halls of Odin, to ask for Odin’s advice. He who sees all would surely know of this theft. He would know if the thief stood among the gods, or if he had fallen to Suttung's wrath.”
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It's not clear what happened to Suttung, I'd assume Odin evaded and lost him, hence why he dropped some of the mead in his haste.
The next day though, the giants came asking questions. Odin sees all in the Norse Cosmos, with his trusted ravens bringing him news every morning. The giants enquired about the theft of their mead, asking if the thief was among the Æsir, or if he had been found and slain by Suttung.
Odin swore an oath to them that he knew nothing of this atrocity, and thus they left, deceived, robbed and outwitted.
Stanza 110
“Odin swore an oath to those giants, that he knew not of this treacherous act. But who could trust Odin? He left Suttung deceived and robbed, and he had broken the heart of Gunnloð.”
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But who can trust Odin? It's very clear that Odin is a maniacal, manipulative and trustworthy individual, often deceiving others to serve his own personal gain.
None the wiser, the giants left to return to Jotunheim, while Odin relished in his successful venture. He gifted the mead of poetry to those he deemed worthy, those great that he deemed worthy of such a prize. And such was the mead of Poetry.