Animal sacrifice done on the day called Eid Al Azha (Bakri Eid) that many so-called Muslims celebrate is like many others acts an innovation falsely attributed to God. Sacrifice is required to be done only by the pilgrims during Hajj or if someone made an intention to do hajj but could not do it for any reasons. There is no requirement on those not doing Hajj to sacrifice any animal, as is traditionally done. This is an innovation.
[2:196] *You shall observe the complete rites of Hajj and`Umrah for GOD. If you are prevented, you shall send an offering, and do not resume cutting your hair until your offering has reached its destination. If you are ill, or suffering a head injury (and you must cut your hair), you shall expiate by fasting, or giving to charity, or some other form of worship. During the normal Hajj, if you break the state of Ihraam (sanctity) between `Umrah and Hajj, you shall expiate by offering an animal sacrifice. If you cannot afford it, you shall fast three days during Hajj and seven when you return home - this completes ten - provided you do not live at the Sacred Masjid. You shall observe GOD, and know that GOD is strict in enforcing retribution. *2:196 See the details of Hajj and `Umrah in Appendix 15.
The other reasons where animal sacrifice is in the order is when the state of Ihraam is violated between Umrah and Hajj and when a game animal is killed during pilgrimage.
[5:95] O you who believe, do not kill any game during pilgrimage. Anyone who kills any game on purpose, his fine shall be a number of livestock animals that is equivalent to the game animals he killed. The judgment shall be set by two equitable people among you. They shall make sure that the offerings reach the Ka`bah. Otherwise, he may expiate by feeding poor people, or by an equivalent fast to atone for his offense. GOD has pardoned past offenses. But if anyone returns to such an offense, GOD will avenge it. GOD is Almighty, Avenger.
Some history from Quran on the animal sacrifice done during Hajj:
Abraham was allowed to be tested by Satan, of course by God's leave, whereby Abraham mistook a dream from Satan to be from God.
Satanic Dream* [37:102-103] When he grew enough to work with him, he said, "My son, I see in a dream that I am sacrificing you. What do you think?" He said, "O my father, do what you are commanded to do. You will find me, GOD willing, patient." They both submitted, and he put his forehead down (to sacrifice him). *37:102 The Most Merciful never advocates evil (7:28). As with Job, Satan claimed that Abraham loved his son too much, and was permitted to put Abraham to that severe test.
We understand from the aforesaid verses that Abraham believed the dream commanding him to sacrifice his son to be from God, when in fact it was from Satan as God does not advocate evil. We also know that Abraham has been cited in the Quran as an exemplary vanguard in submission to God alone.
[16:120] Abraham was indeed an exemplary vanguard in his submission to GOD, a monotheist who never worshiped idols.
He is also known as God's chosen friend in 4:125. Since Abraham was a submitter to God alone, God intervened and protected Abraham and Ismail from such a heinous act.
[37:104] We called him: "O Abraham.
[37:105] "You have believed the dream." We thus reward the righteous.
[37:106] That was an exacting test indeed.
[37:107] We ransomed (Ismail) by substituting an animal sacrifice.
[37:108] And we preserved his history for subsequent generations.
As we see above, God willed to preserve this history of Abraham and Ismail. It demonstrates their absolute submission to God alone. And this preservation is done through the ritual of animal sacrifice during Hajj.
[22:36] The animal offerings are among the rites decreed by GOD for your own good.* You shall mention GOD's name on them while they are standing in line. Once they are offered for sacrifice, you shall eat therefrom and feed the poor and the needy. This is why we subdued them for you, that you may show your appreciation.