Dear dagn,
eng789 is correct.
Oxford Dictionary of English
"Draw blood" = "Cause someone to bleed, especially in the course of a fight".
Example: "The blow drew blood from the corner of his mouth".
The writer has long fingernails and could have written this:
"I grip my neck, like a bird with long claws, and I begin to claw my neck so hard that my fingernails enter my skin. Blood begins to come from the deep scratch, the wound.
I draw blood. I make myself bleed by scratching my neck very hard".
I hope that this helps you.
Les