Muslims slaughter animals during `Eid to emulate Prophet Abraham who was about to slaughter his son, Isma`eel (Ishmael), to show obedience to Almighty Allah, when Ishmael was redeemed by a ram brought to Abraham by Angel Gabriel. The Angel then conveyed to him Allah’s Command to slaughter the ram instead.
With regard to what to say when slaughtering udhiyah, the late Sheikh Ibn `Uthaymeen, a prominent Saudi Muslim scholar, said:
The sunnah for one who wants to slaughter the udhiyah (sacrifice) is to say this when slaughtering it:
“Bismillaah, wa Allahu akbar, Allahumma haadha minka wa laka, haadha ‘anni. Allahumma taqabbal minni wa min aali.” (In the name of Allah, Allah is most great. O Allah, this is from You and to You. This is on my behalf. O Allah, accept it from me and from my family.)
Or if it is being offered on behalf of someone else, say this:
“Bismillaah, wa Allahu akbar, Allahumma haadha minka wa laka, haadha ‘an (the person’s name), Allahumma taqabbal min (the person’s name), wa aali (the person’s name).” (In the name of Allah, Allah is most great. O Allah, this is from You and to You. This is on behalf of the person’s name. O Allah, accept this from the person’s name and from the family of the person’s name.)
What is obligatory here is to say Bismillaah; the rest is mustahabb (recommended) but is not obligatory.
Al-Bukhari and Muslim narrated that Anas said: The Prophet (peace and blessings be upon him) sacrificed two horned rams that were white speckled with black. He slaughtered them with his own hand, said “Bismillaah” and “Allahu akbar,” and put his foot on their necks.
Muslim narrated from `A’ishah that the Messenger of Allah (peace and blessings be upon him) ordered that a horned ram be brought to him so that he could sacrifice it. He said, “`A’ishah, give me the knife.” Then he said, “Sharpen it on a stone.” So she did that, then he took it and took the ram, and he laid it down and prepared to slaughter it. He said, “In the name of Allah, O Allah accept (this sacrifice) from Muhammad and the family of Muhammad and the Ummah of Muhammad,” then he sacrificed it. (Musnad Ahmad)
At-Tirmidhi narrated that Jabir ibn `Abdullah said: I was present with the Prophet (peace and blessings be upon him) on the Day of Sacrifice at the prayer place. When he had finished his sermon he came down from his pulpit and a ram was brought, which the Messenger of Allah (peace and blessings be upon him) slaughtered with his own hand. He said, “In the name of Allah, Allah is Most Great. This is on behalf of myself and on behalf of those of my Ummah who have not offered a sacrifice.” ( Abu Dawud and al-Tirmidhi). In some reports the words “O Allah, this is from You and to You” are added.
Allahumma minka (O Allah, this is from You) means “this sacrifice is a gift and provision that has reached me from You.” Laka (to You) means “it is sincerely for You alone.”