What exactly is the Torah teaching us here? What is the reason for the Isur of drinking blood?
Rashi: It is teaching us why Hashem deals so strictly with the person who drinks blood - because the living Soul of all creatures lies in the blood. 1
Ramban #2: When Hashem created the world for the benefit of mankind, He only permitted them to eat what grows from the ground 4 - not animals with a Soul; and it was only when the latter were saved on the merit of No'ach and No'ach brought some of the animals on the Mizbe'ach in honor of Hashem that He allowed man to eat them. 5 However, it is only their bodies (which He initially created for his benefit) that He permitted; their Souls, 6 He designated to atone for man's Soul on the Mizbe'ach. 7
Ramban #3: What a person eats becomes part and parcel of the eater. Consequently, if he drinks the blood of an animal, it mixes with his blood and imbues him with the coarse qualities of the animal whose blood he drank. 8
Ramban (citing Rambam in Moreh Nevuchim): The Torah forbids drinking blood, and is Metaher and Mechaper through sprinkling it and putting it on the Mizbe'ach, to counter the Kasdim (Babylonians) who despised blood, but would eat it to connect to Sheidim to find out the future. 9
Rashi: Which explains why the blood of Korbanos is sprinkled on the Mizbe'ach to atone for man's sins (Nefesh be'Nefesh), and not because Hashem (Kevayachol) is in any way partial towards it (Seforno).
And the reason that the blood of Chayos is included in the prohibition is because, seeing as the two bloods are indistinguishable (as opposed to the Cheilav of a Chayah, which looks different than that of a Beheimah and which is therefore permitted), if the blood of Chayos were to be permitted, one would come to permit the blood of a Beheimah as well (Ramban).
For the same reason that Cheilev is forbidden (Ramban).
See Ramban (DH 've'Al ha'Derech'), who discusses the spirituality of an animal's Soul.
Ramban DH Min ha'Yadu'a Od: But not to eat, as it is not appropriate for one Soul to eat another Soul, since all Souls belong to Hashem (See Koheles, 3:19 [Ibid.]). It is also inappropriate for the Soul of man - which goes up (because it is infinite) - to combine with that of an animal - which goes down (because it is finite - See Koheles, 3:21).
Which is not the case when he eats the animal's meat, since it changes its format when he chews it (Ramban).
Ramban (DH veha'Rav): This is a nice explanation, but that it is unlike the Pesukim, which forbid 'because the blood is the Nefesh.'