The following are quotes of Srila Prabhupada regarding Vedic animal sacrifices:
Lecture on BG 7.16 -- Bombay, April 7, 1971:
Prabhupāda: There are tāmasika-purāṇas where it is recommended that if you want to eat flesh, then you can get a goat and sacrifice before Goddess Kālī and you can eat that. The purpose is that if a conditioned soul has got the natural tendencies, then why they are mentioned in the śāstras? The idea is... Just like Lord Buddha. Lord Buddha, his mission was to stop animal killing. Ahiṁsā paramo dharmaḥ. Lord Buddha appeared, being compassionate with the poor animals. Sadaya-hṛdaya darśita-paśu-ghātam. There is a description of Lord Buddha's activities by a Vaiṣṇava poet, Jayadeva Gosvāmī. He says, praying to Lord Buddha, nindasi yajña-vidher ahaha śruti-jātaṁ sadaya-hṛdaya darśita-paśu-ghātam keśava dhṛta-buddha-śarīra jaya jagadīśa hare Lord Buddha appeared to stop this animal killing. But because in the Purāṇas there are sometimes regulative principle of killing animal, therefore he had to deny the authority of Vedas, because those who are after killing animals, they will find some clue that "Here in the Vedas the animal-killing is sometimes recommended." But that animal-killing is not a, I mean to say, instigation that "You go on killing animals." You can understand by a nice example. Just like the government opens liquor shop. It does not mean the government is encouraging to drink liquor. It is not like that. The idea is that if government does not allow some drunkards to drink, they will create havoc. They will distill illicit distillation of liquor. To check them, the government opens liquor shop with very, very great, high price. The cost... If the cost is one rupee, government excise department charges sixty rupees. So the idea is not to encourage, but to restrict. The idea is prohibition, at least in our country. Similarly, when there is allowance for sex life or meat-eating or drinking in the śāstras, they are not meant for instigating that "You go on with this business as much as you can." No. Actually they are meant for restriction.
Lecture on SB 1.5.14 -- New Vrindaban, June 18, 1969:
Prabhupāda: According to Vedic śāstra, meat-eating is not allowed by purchasing from the slaughterhouse. No. They... There is motive. The marriage ceremony or the meat-eating, the so much ritualistic performances, there is motive. What is that motive? Motive is restriction. Just like the same example, marriage: the real idea is to restrict the boy and the girl to one woman and one man. That is the idea, main idea. If he's not married, then he will be just like cats and dogs. So idea is very good. But Nārada Muni says, "After all, you're coming to the point of sex life. So why so much propaganda?" Similarly, for meat-eating, there is also sanction in the śāstras, tāmasika-śāstra, not sāttvika. There are three divisions of śāstra-sāttvika, rājasika and tāmasika. Those who are meat-eaters, cannot give up meat-eating, for them, the prescription is, "All right, you can eat meat, but you have to sacrifice one goat, but not cow." The... For sacrifice the animal is recommended: goat. So you can, I mean to say, cut throat of a goat in the presence of goddess Kālī and you can eat. There are so many prescriptions. But that is also restriction, that Kālī-worship is one day in a month, on the dark moon day. What is called, dark moon? Amāvasyā. Full moon and...? Eh? No, no, no. When there is... On that particular..., there is no moon in the sky.
Hayagrīva(?): The dark of the moon.
Prabhupāda: It is called dark moon, yes. So that Kālī-pūjā is recommended on the dark moon day. The... That is one day in a month. Similarly, according to marriage life, the sex life is also allowed one day in a month. The whole thing is restriction. Similarly, drinking wine also, there is Devī-pūjā. That is also once in a year or something like that. The whole point is restriction.
As we can see Prabhupada claims that animal sacrifices made to Goddess Kali are solely for the purpose for restricting gratifications of meat eaters. However, many people have told me that animals sacrificed to Mother Kali is for rituals and not for one's own gratification. Are both groups correct?