Emborracharme literally means "to get myself drunk". While it often refers to alcohol, its use here is purely metaphorical and much more romantic.
In the song, Morat sings "Emborracharme así, de ti" (To get drunk like this, on you). He uses this powerful verb to describe the intoxicating feeling of being completely overwhelmed by love, as if the person he's with is the only "drink" he needs. It's a beautiful, poetic twist on a common word.
Have you ever wondered if love at first sight is real? Morat makes a strong case for it in this song. It's a story about a completely spontaneous and successful romantic encounter. The singer sees someone in Madrid during the summer and is instantly smitten, comparing his thirst for them to being drunk: "Beber de ti, de ti, de ti / Emborracharme así, de ti" (Drink from you, from you, from you / Get drunk like this, on you).
He then makes an incredibly bold move. As a total stranger, he walks up and says, "te dije te amo / Te he estado buscando por más de mil años" (I told you I love you / I've been looking for you for more than a thousand years). Instead of running away, the other person simply asks, "¿a dónde vamos?" (where are we going?). That one question kicks off their entire relationship. They beat the odds and are still together, describing themselves as "la prueba de que existe amor a primera vista" (the proof that love at first sight exists).