In this context, "grate" means that the sound of Al sobbing is causing discomfort or irritation, as if the sound is harsh or unpleasant. It's a figurative use of the word to convey the emotional impact of Al's sobbing on the narrator.
It means that the sound of Al sobbing is so distressing or unsettling that it feels uncomfortable or painful for the narrator. The word "grate" is used to describe the emotional reaction to the sound, as if it's causing a sharp or unpleasant sensation in the narrator's throat.