Anestesiar means "to anesthetize" or "to numb." It's a striking word that isn't commonly found in everyday song lyrics, making it stand out.
In the context of "Buscando El Olvido," the singer uses it to describe his attempt to numb the pain of a breakup with alcohol. He sings, "Se toma de más para anestesiar este dolor," vividly illustrating his desire to escape the emotional agony, which makes the word particularly evocative and memorable.
"Buscando El Olvido" ("Searching For Oblivion") is a heartbreak song about drinking to forget a lost love. Adriel Favela opens with the same bottles and the same friends, but tonight the mood has changed: he isn't out to celebrate, he's out to numb the pain.
Through the song he keeps circling back to one idea: no amount of good sense can talk him out of drinking more when he's hurting. Each round of drinks is another attempt to stop thinking about her, to blur the memory of her kisses, even for a few hours.
By the end the singer admits the truth he can't drink away: she isn't coming back. The drinking isn't really working either, since he still remembers exactly how much he loved her. It's a song about the gap between wanting to forget and actually being able to.