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 enimals 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
Which explains why the blood of Korbanos is sprinkled on the Mizbe'ach to atone for man's sins (Nefesh be'Nefesh [Rashi, Seforno and Targum Yonasan]), 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). The Rambam (in Moreh Nevuchim) attributes the Isur of drinking blood on the one hand and the Mitzvah to sacrificing it on the other, to the Kasdim (the Babylonians) who despised blood, but who would use it to attract the Sheidim to find out the future. So Hashem counters their foolishness by instructing us to do the opposite (See Ramban DH 've'ha'Rav', who praises the Rambam's explanation, but adds that it does not go withy the Pesukim).
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.
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 [Ramban DH 'min ha'Yadua' Od']).
Which is not the case when he eats the animal's meat, since it changes its format when he chews it (Ramban).