Why did the blood sacrifice in Genesis take place as acceptable to God?

From the beginning, the requirement of a sacrifice as a substitute for their sins was communicated to them verbally. It is not found directly in Genesis, but strongly inferred in Genesis 4 when Abel's animal sacrifice was accepted over Caine's offering of fruits. The act of offering an animal sacrifice was an acknowledgement of their need for a savior, for by their own merits they can never atone for sin.

And in the process of time it came to pass that Cain brought an offering of the fruit of the ground to the Lord. Abel also brought of the firstborn of his flock and of their fat. And the Lord respected Abel and his offering, but He did not respect Cain and his offering. (Genesis 4:3-5)

The next verse implies that Cain knew what was the right way but chose not to do it.

So the Lord said to Cain, Why are you angry? And why has your countenance fallen? If you do well, will you not be accepted?" (Genesis 4:6-7)

Paul also elaborate further on this event:

By faith Abel offered to God a more excellent sacrifice than Cain, through which he obtained witness that he was righteous, God testifying of his gifts; and through it he being dead still speaks. (Hebrews 11:4)

In addition to Genesis 4, we has have evidence of the very first animal sacrifice. An animal was slain to make clothes for Adam and Eve in place of their covering of leaves.

"Also for Adam and his wife the Lord God made tunics of skin, and clothed them." (Genesis 3:21)

We are to take off our own filthy robes and put on the robe of Christ (Zech 3:4)

In Genesis God told Adam that:

Genesis 2:17 NKJV but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die."

In the original Hebrew the word used here is:

תָּמֽוּת׃ tā·mūṯ.

you will die

And if we break that down even farther, according to Brown Driver and Briggs:

מוּת mûth

BDB Definition:

1) to die, kill, have one executed

1a) (Qal)

1a1) to die

1a2) to die (as penalty), be put to death

1a3) to die, perish (of a nation)

1a4) to die prematurely (by neglect of wise moral conduct)

1b) (Polel) to kill, put to death, dispatch

1c) (Hiphil) to kill, put to death

1d) (Hophal)

1d1) to be killed, be put to death

1d1a) to die prematurely

Part of Speech: verb

A Related Word by BDB/Strong’s Number: a primitive root

To die is bring life to an end, and God forbad man to eat Blood because:

Genesis 9:2 through 6 NKJV And the fear of you and the dread of you shall be on every beast of the earth, on every bird of the air, on all that move on the earth, and on all the fish of the sea. They are given into your hand. 3 Every moving thing that lives shall be food for you. I have given you all things, even as the green herbs. 4 But you shall not eat flesh with its life, that is, its blood. 5 Surely for your lifeblood I will demand a reckoning; from the hand of every beast I will require it, and from the hand of man. From the hand of every man's brother I will require the life of man. 6 "Whoever sheds man's blood, By man his blood shall be shed; For in the image of God He made man.

In Leviticus God allowed the substitution of the life of animals as a substitute for the life of man in the letting of the animals blood as stated in Surely for your lifeblood I will demand a reckoning; from the hand of every beast I will require it,

In the same manner as he allowed the blood (life) of Jesus to atone for our sins.

The sacrificial system instituted by God in the Old Testament is a prevue to the sacrifice of Jesus.

Just as the blood of the animals is only representative of forgiveness, it was of no value without repentance, So is the blood of Christ only representative of our forgiveness without repentance.

From the very beginning God declared that sin can only be washed in blood.

Either our sins will be washed in the blood of Christ or it will be washed in our own blood:

Revelation 20:14 and 15 NKJV Then Death and Hades were cast into the lake of fire. This is the second death. 15 And anyone not found written in the Book of Life was cast into the lake of fire.

In the Pearl of Great Price in the book of Moses it says:(Moses 5:5-7)

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.

When Adam left the Garden of Eden he was commanded by God to give a sacrifice. We see this is continued on by his sons with the whole Cain and Abel situation. It was meant as a similitude of Christs sacrifice. So when Christ died on the cross that law was fulfilled and it was no longer needed for man to sacrifice living animals though Christ has asked other sacrifices of man.

I don't think there is anything written in Genesis specifically if you look at it chronologically but if you look at the story of Abraham and his son Isaac Genesis talks about sacrifice there. But it doesn't explain anything or imply whether sacrifice had been happening before. It is an interesting story as Abraham was almost sacrificed by his own father and that was the reason he left his homeland, so for him to do the same to his only son was very significant.

It was probably written in other scriptures that have not survived to this time era, which is unfortunate as that would have been interesting.