Its when the baby is left attached to the placenta by the umbilical cord, untill it naturally detaches. Im not 100% on all the supposed benefits of this however I guess it would be similar to delayed cord clamping, when they wait for the cord to stop pulsing, giving baby an increased blood flow / nutrients from the placenta but leave it all to nature in terms of drying up and detaching . Id image there must be quite a high risk of infection with it even in "normal" circumstances.
You dont have to have the injection to make the placenta come way, its just something they do to speed up thr process, and you can also retain placenta even if you do have the injection too, even with a section it can happen.
Burying the placenta is a tradition of some religions, paganism is one especially the bury ubder a tree part. Its actually quite popular now to dry the placenta and make capsules from it, the health benefits of it are quite something, including increased milk supply. If you think in nature an animal will eat the placenta.