Rajo comes from the verb rajarse, which literally means "to tear" or "to split." However, in Mexican slang, it commonly means "to back down," "to give up," or "to chicken out."
In "Ya Lloré," the singer declares, "Pero no me rajo" (But I won't give up/back down), despite the pain of infidelity. This phrase adds a defiant and resilient tone to the song, showing the singer's determination even in heartbreak.
"Ya Lloré" is a banda ballad about heartbreak and betrayal. The singer has already done his crying, and now he's trying to convince himself it isn't worth crying over anymore.
He stayed faithful, but his partner left him for someone with more money and more polish. He knows he can't compete with that, and it stings, but he still refuses to give up on himself, even while admitting the pretty promises of love he was given are ones he'll never be able to forget.
By the end he's back to pouring himself a drink and crying anyway. He was the one who stayed loyal, and that's exactly what makes losing her hurt so much.