Rav Natan of Breslav presents a Chasidic explanation of this law that builds on a remarkable number of principles discussed in previous columns.
The main principle at work is the one which we mentioned in the column on ritual slaughter (shechita) as well as on salting meat and the prohibition of meat and milk: kosher animals are those which have a "neutral" aspect, which is susceptible to assimilation into holiness, and a completely bestial nature which can not be. The act of slaughter forcibly subdues the animating bestial nature; the remaining meat is a spiritually inert food, which is then permissible. However, the blood still embodies the original animal nature, for as the following verse reminds us, "for the spirit of all flesh is its blood". The blood of the slaughter is not eaten, and the blood within the meat is salted out.
These steps, slaughter and salting, are sufficient for domestic beasts. But they are not sufficient for game animals and birds. The reason is that the nature of these animals is not quite so bestial as that of domestic species such as cows, goats, and sheep. Making a spiritual distinction among animals is hardly surprising in itself; after all, most species of animals are much more bestial and are not permissible at all.
There are various ways in which we can explain or express the special advantage of game animals and fowl; the most intuitive is to point out that these animals have more freedom and independence than domestic beasts. Wild animals are completely free, while even domestic birds have the wherewithal for freedom in their wings. Our tradition esteems this natural free state; the Yaavetz writes that the reason we have to feed our animals before ourselves is because by domesticating them we have deprived them of their freedom to provide for themselves (Sheilat Yaavetz I:17). More precisely, wild animals obtain their sustenance directly from the Creator, without human intervention (See Rashi on Bereshit 8:11).
Consequently, the blood of these creatures doesn't embody a pure bestial nature but rather contains certain "sparks" of a higher, quasi-human nature, as our freedom and independence are salient features of our humanity.
The presence of these sparks of a higher nature in the blood, which is the spirit, of these beings, means that this blood requires interment (burial), just as a human body requires interment due to the fact that a bit of its previous human nature continues to inhere in it.
To elaborate further, as we explained in the column on burial, interment dignifies because it covers the shame of the body which was once the abode of the holy soul and is now reduced to a cadaver, and atones because once the body returns to the earth, it can then serve as the raw material for new life, as man was originally created from the dust of the earth. By the same token, "burying" the blood covers up the indignity of the still-dominant bestial nature of the blood, and atones by enabling the few remaining holy sparks to separate themselves and return to their source in holiness. - (Based on Likutei Halakhot Shechita 2:8)