Why do so many religions require sacrifice of animals? What does the explanation imply for understanding the sacrifice of Christ?

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I used to read great swathes of stuff about comparative religions and mythology; and they all assumed that the sacrificing of animals was something that human did naturally - as a universal aspect of human psychology.

But this never really convinced me, because I did not feel this impulse in myself - nor did I see it in others. I did feel and see an near universal impulse among humans to torture other creatures including other humans, but not to sacrifice them.

I have just come across an interesting passage in James E Talmage's book Jesus The Christ (1915) in his notes for Chapter 5:

NOTES TO CHAPTER 5.

1. The Antiquity of Sacrifice as a Prototype of Christ's Atoning Death.—While the Biblical record expressly attests the offering of sacrifices long prior to Israel's exodus from Egypt—e.g. by Abel and by Cain (Gen. 4:3, 4); by Noah after the deluge (Gen. 8:20); by Abraham (Gen. 22:2, 13); by Jacob (Gen. 31:54; 46:1)—it is silent concerning the divine origin of sacrifice as a propitiatory requirement prefiguring the atoning death of Jesus Christ. The difficulty of determining time and circumstance, under which the offering of symbolical sacrifices originated amongst mankind, is recognized by all investigators save those who admit the validity of modern revelation. The necessity of assuming early instruction from God to man on the subject has been asserted by many Bible scholars. Thus, the writer of the article "Sacrifice" in the Cassell Bible Dictionary says: "The idea of sacrifice is prominent throughout the scriptures, and one of the most ancient and widely recognized in the rites of religion throughout the world. There is also a remarkable similarity in the developments and applications of the idea. On these and other accounts it has been judiciously inferred that sacrifice formed an element in the primeval worship of man; and that its universality is not merely an indirect argument for the unity of the human race, but an illustration and confirmation of the first inspired pages of the world's history. The notion of sacrifice can hardly be viewed as a product of unassisted human nature, and must therefore be traced to a higher source and viewed as a divine revelation to primitive man."

Smith's Dic. of the Bible presents the following: "In tracing the history of sacrifice from its first beginning to its perfect development in the Mosaic ritual, we are at once met by the long-disputed question as to the origin of sacrifice, whether it arose from a natural instinct of man, sanctioned and guided by God, or was the subject of some distinct primeval revelation. There can be no doubt that sacrifice was sanctioned by God's Law, with a special, typical reference to the Atonement of Christ; its universal prevalence, independent of, and often opposed to, man's natural reasonings on his relation to God, shows it to have been primeval, and deeply rooted in the instincts of humanity. Whether it was first enjoined by an external command, or was based on that sense of sin and lost communion with God, which is stamped by His hand on the heart of man—is an historical question, perhaps insoluble."

The difficulty vanishes, and the "historical question" as to the origin of sacrifice is definitely solved by the revelations of God in the current dispensation, whereby parts of the record of Moses—not contained in the Bible—have been restored to human knowledge. The scripture quoted in the text (pp. 43, 44) makes clear the fact that the offering of sacrifices was required of Adam after his transgression, and that the significance of the divinely established requirement was explained in fulness to the patriarch of the race. The shedding of the blood of animals in sacrifice[Pg 54] to God, as a prototype "of the sacrifice of the Only Begotten of the Father," dates from the time immediately following the fall. Its origin is based on a specific revelation to Adam. See P. of G.P., Moses 5:5-8.

http://www.gutenberg.org/files/22542/22542-h/22542-h.htm#chapter_5

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The final reference is to The Pearl of Great Price, and it reads:

 5 And he gave unto them commandments, that they should worship the Lord their God, and should offer the firstlings of their flocks, for an offering unto the Lord. And Adam was obedient unto the commandments of the Lord.

 6 And after many days an angel of the Lord appeared unto Adam, saying: Why dost thou offer sacrifices unto the Lord? And Adam said unto him: I know not, save the Lord commanded me.

 7 And then the angel spake, saying: This thing is a similitude of the sacrifice of the Only Begotten of the Father, which is full of grace and truth.

 8 Wherefore, thou shalt do all that thou doest in the name of the Son, and thou shalt repent and call upon God in the name of the Son forevermore.

https://www.lds.org/scriptures/pgp/moses/5?lang=eng

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I find this explanation - that, far from being a spontaneous product of human psychology; the practice of sacrifice was instead an instruction of divine revelation, given to many men in many places throughout history as a 'similitude' of Christ's sacrifice - to be immediately intriguing, and on reflection psychologically convincing. 

One reason I find it convincing is that this is how Christ's sacrifice seemed to work in history. In ancient societies where sacrifice was performed, there was an immediate understanding of the meaning of Christ's death - and such societies were readily convertible to Christianity. 

But modern secular societies, which lack the institution of religious sacrifice, we are often puzzled by the meaning, and sometimes even repelled by the notion, of Christ as a sacrifice. 

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