Normal feathers have hollow shafts but solid structures in the vane. Your griffons have hollow channels along the outer surface of the barbs of their feathers that are coated internally with luciferin-producing cells and luciferase-producing nodes. When combined with oxygen in the atmosphere this produces light for a brief period until the reagents are exhausted.
In the rest state the majority of the tubes are filled with oxygen-depleted air. In order to activate the light emission a small amount of fresh air is pushed into the base of the feather by a modified arrector pili muscle. This provides fresh oxygen that then reacts to form a pulse of light. The griffon can control this to a certain degree, tensing areas of skin to raise and briefly illuminate the feathers for aggression or mating displays.
The same chemical process is used for more permanently illuminated areas such as the tail, only the oxygen supply mechanism differs. Rather than tightening the skin to produce light these areas are structured so that they can be opened to the air directly to produce low levels of light, or closed to conserve resources.
The downside of this adaptation is that the structure of the feathers is no longer able to encourage proper laminar airflow, vastly reducing their aerodynamic effectiveness. Over time this resulted in the griffons losing their flight ability, leading to a variety of evolutionary changes: redistribution of flight muscle mass, smaller wings that don't impede movement through trees and such, all of the long feathers have been replaced with short ones, wing structures are much too light to support much weight, etc.
Only healthy, well-fed griffons with good genetics are able to sustain the light production for more than a few seconds at a time. A more impressive and lengthy light show is a demonstration to potential mates that the griffon is in peak health and is a prime candidate for breeding. If there is competition in the area the males will fight not to drive the other away but to damage their feathers, reducing their ability to put on a good mating display.
As to diet, there's no reason why all griffons can't be omnivores just like some birds are. In fact some of the necessary nutrients to support their production of luciferin comes from certain types of plant. Access to these plants is a major resource requirement for your griffons to occupy an area. Without them the griffons can't make light for their mating dances, which will fairly quickly result in their population falling below sustainable levels.
Of course your humans have no idea about all of this because, well, they're usually too busy trying to avoid the sharp bits. And it's a real shame because what the humans haven't quite figured out is that some of their staple crops are execellent sources of the required nutrients. That's why humans have so many problems with griffons terrorising their farming communities. If they'd just put in the research to figure it out instead of sending hordes of murder hobos in to kill everything, maybe they'd realise that if they'd just quit planting all that blue corn everywhere then maybe they'd save a few gold on hunters.
Maybe then we could get on with breeding domestic house-griffons that could replace felis catus as the pet of choice for keeping rats out of the kitchen.