Aferrarme comes from the verb aferrar, meaning 'to cling to', 'to hold on tightly', or 'to grasp'. It often implies a strong, sometimes desperate, attachment.
In the context of "Estos Tragos Van Por Ti," the singer says, "Mas no debería aferrarme / Porque sé que te perdí" (But I shouldn't cling / Because I know I lost you). This reveals his internal struggle: despite his pain and desire, he knows he must let go, making aferrarme a poignant and relatable word for anyone who has struggled to move on from a past love.
Carín León opens by breaking a promise he makes in the same breath: "Yo que juré que nunca tomaría" [I, who swore I would never drink], and two lines later "Ya llevo varios tequilas" [I've already had several tequilas]. He blames the tequila on her, which is exactly the drinker's logic. "Es por culpa de tus besos / Que aún no olvidé" [It's because of your kisses that I still haven't forgotten] puts the fault on a mouth he can't stop remembering, never on the hand holding the glass.
Then the drinking stops working. "Y entre más sigo tomando / Los recuerdos van llegando" [And the more I keep drinking, the memories keep coming], and instead of blurring her they sharpen: "Pero me están torturando" [But they're torturing me]. The real fight in this song is not with her at all. "El orgullo está peleando / Con mis ganas de llamarte" [Pride is fighting with my urge to call you], and pride wins. The phone stays on the table, and the toast he raises instead, "Y estos tragos van por ti" [And these drinks are for you], is as close as he lets himself get to dialling.