Blood is life, both literally and figuratively.
The result of sin is death, the opposite of life.
Adam and Eve sinned (they disobeyed God), they believed the Serpent's lie that they would not "surely die", and as a result they earned death.
Blood sacrifices symbolize this relationship: only death (shed blood) can atone for sin.
Consider the offerings of Cain and Abel:
… 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.
— Genesis 4:3–4
… the LORD respected Abel and his offering,
but He did not respect Cain and his offering. …
— Genesis 4:4–5
Cain sacrificed a living breathing soul (נֶפֶשׁ, nep̄eš) to God, while Abel offered some vegetables.
Cain too could have offered a lamb, simply by trading some of his crops with his brother, but he didn't.
He personally gave up just as much as Abel, but he failed to understand the true meaning of blood sacrifices.
For God, it isn't one's intentions that count; it is one's actual deeds.
Good intentions aren't good enough.
The Christian scriptures make this concept quite clear:
Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?
And then will I profess unto them, I never knew you: depart from me, ye that work iniquity.
— Matthew 7:21–23
Noah was reminded of the significance of blood:
But you shall not eat flesh with its life, that is, its blood.
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.
Whoever sheds man’s blood,
By man his blood shall be shed;
For in the image of God
He made man.
— Genesis 9:4–6
This is considered one of the Noahide Laws, applicable to all mankind:
And whatever man of the house of Israel, or of the strangers who dwell among you, who eats any blood, I will set My face against that person who eats blood, and will cut him off from among his people.
‘For the life [soul] of the flesh is in the blood, and I have given it to you upon the altar to make atonement for your souls; for it is the blood that makes atonement for the soul.’
Therefore I said to the children of Israel, ‘No one among you shall eat blood, nor shall any stranger who dwells among you eat blood.’
— Leviticus 17:10–12
Only be sure that you do not eat the blood, for the blood is the life; you may not eat the life with the meat.
You shall not eat it; you shall pour it on the earth like water.
You shall not eat it, that it may go well with you and your children after you, when you do what is right in the sight of the LORD.
— Deuteronomy 12:23–25
Blood must be returned to the Earth, symbolically returning life to the God that created it.
Look at the original Passover:
Your lamb shall be without blemish, a male of the first year. You may take it from the sheep or from the goats.
Now you shall keep it until the fourteenth day of the same month. Then the whole assembly of the congregation of Israel shall kill it at twilight.
And they shall take some of the blood and put it on the two doorposts and on the lintel of the houses where they eat it.
Now the blood shall be a sign for you on the houses where you are. And when I see the blood, I will pass over you; and the plague shall not be on you to destroy you when I strike the land of Egypt.
— Exodus 12:5–13
The blood of the sacrificial lamb symbolized life, and offered salvation from certain death to those that made use of it.
The word "sacrifice" occurs hundreds of times in the Bible, almost all of them referring to the literal killing, by bleeding, of a living breathing soul.
And you shall offer your burnt offerings, the meat and the blood, on the altar of the LORD your God; and the blood of your sacrifices shall be poured out on the altar of the LORD your God, and you shall eat the meat.
— Deuteronomy 12:27
These sacrifices act not only as as an admission of one's sins, but acknowledgement that all life belongs to God, and that sin (disobeying God) will ultimately require returning one's own life to him:
Behold, all souls are Mine;
The soul of the father
As well as the soul of the son is Mine;
The soul who sins shall die.
— Ezekiel 18:4
The Talmud (Jewish commentary and records of oral tradition) makes frequent references to the fact that "there is no atonement without blood".
From a Christian perspective, Jesus shed his blood (i.e. died) as the ultimate sacrifice, to pay the death penalty for anyone that asks:
For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord.
— Romans 6:23
That the sacrificial animals had to be "without blemish" etc. symbolizes Jesus's own perfect life.
All previous animal sacrifices merely served as symbolic foreshadowing of this single momentous event.