I was wondering if someone would ask this! Good question.
“Graf” is short for paragraph in journalism lingo. It can even be used alone to refer to any paragraph.
“Nut” refers to the meat, the kernel, the nut of the story–what it’s about. Poynter’s Chip Scanlan writes, “It’s called the nut graf because, like a nut, it contains the ‘kernel,’ or essential theme, of the story.”