Embriagándome comes from the verb embriagar, which literally means "to get drunk" or "to intoxicate". It's a powerful and poetic word that you won't hear every day.
In the song, Natalia Lafourcade sings Embriagándome en su aroma (Intoxicating myself with its aroma) to describe being completely overwhelmed by the scent of the sea. She's using the word metaphorically to express a feeling of being blissfully lost and enveloped in the moment, as if she's drunk on the ocean itself.
This beautiful song by Mexican artist Natalia Lafourcade feels like a conversation with the ocean itself. The singer is moving on from something in her past, enjoying a “dulce adiós” (sweet goodbye) and greeting her present moment. She finds signs of a new person or a new beginning in the sounds of the sea, like in the line “Por un canto de gaviotas / Supe que allí estabas tú” (Because of a song of seagulls / I knew you were there).
But at the heart of the song is a deep feeling of solitude. Instead of running from it, she embraces it. She asks the sea to sing to her, specifically “Un bolero de soledad” (a bolero of loneliness), a slow, romantic song for her solitary state. She says she walks “sola, con soledad” (alone, with loneliness), treating her solitude as a companion. The song ends by simply naming its two main characters: “Soledad y el mar” (Loneliness and the sea).