When a new feather is growing in, it has a blood supply in the shaft to feed it while it grows. Feathers with this blood supply are called blood feathers. After the feather is fully grown, the blood recedes, and the feather is no longer a blood feather.
Birds break blood feathers all the time. Most commonly, it happens in the wings and in the tail, because those are the places most likely to get banged up against something. When a blood feather breaks, it bleeds. The bleeding sometimes stops on it's own, but if the feather is not pulled it can get bumped again and start bleeding again.
For pet birds, it is usually recommended that the feather gets pulled so that it cannot bleed again.
Interestingly enough, if you have a bird that you want to fly, you generally don't want to pull the feather. Rehabbers and falconers will sometimes try to splint the feather to it's neighbor so it can't get bumped. If the feather can be left alone long enough for it to finish growing, the blood supply will recede and you won't have a problem anymore.
But for pet birds who don't need to fly to hunt to stay alive, who could easily bleed to death if a blood feather was bumped and injured in the middle of the night, it's much safer just to pull it. This stimulates a new feather to grow behind it.