The Meaning of the Word Atone
The first usage of the word atone is the Bible is found in Genesis 6:14 and is translated as the verb cover, coat, or pitch depending on the translation:
Make yourself an ark of gopher wood. Make rooms in the ark, and
cover it inside and out with pitch. (ESV)
So make yourself an ark of cypress wood; make rooms in it and coat it
with pitch inside and out. (NIV)
Make thee an ark of gopher wood; rooms shalt thou make in the ark, and
shalt pitch it within and without with pitch. (KJV)
Make thee an ark of gopher wood; with rooms shalt thou make the ark,
and shalt pitch it within and without with pitch. (JPS)
Since this is the first usage, I would argue that this is the literal, prototypical meaning which gives rise to the later, more figurative meaning of reconcile, make amends, or propitiate.
The reason animal sacrifices atone (reconcile, make amends, propitiate) is that they cover the sin of the person being atoned.
This can be seen clearly when we trace the concept of animal sacrifice back to it's original formation. As early as the story of Cain and Abel (Genesis 4) the concept of animal sacrifice sounds like an established institution. This points us back to Genesis 2 and 3 where we find the following narrative arc:
A) Naked an unashamed
This is the initial condition of Adam and Eve before they have sinned.
And the man and his wife were both naked and were not ashamed. (Gen
B) Naked and ashamed
As soon as Adam and Eve disobey and eat the forbidden fruit, they are aware of their shameful nakedness.
Then the eyes of both were opened, and they knew that they were naked.
And they sewed fig leaves together and made themselves loincloths.
(Gen 3:7 ESV)
C) Futile Self Covering
They attempt to cover themselves with fig leaves, but these attempts are futile. They can not cover their shame, are now afraid to see God and hide from him.
And he said, “I heard the sound of you in the garden, and I was
afraid, because I was naked, and I hid myself.” He said, “Who told
you that you were naked? Have you eaten of the tree of which I
commanded you not to eat?” (Gen 3:10,11 ESV)
D) God covers their nakedness
And the LORD God made for Adam and for his wife garments of skins and
clothed them. (Gen 3:21 ESV)
E) Animal Sacrifice
Whereas Adam and Eve's attempts to use fig leaves fails to cover their nakedness, God succeeds by sacrificing animals and using their skins to clothe Adam and Eve.
The symbolism here contrasts Adam and Eve's own methods to cover themselves (from the ground which is cursed because of them) to God's methods (which require a bloody sacrifice).
The Basis of the Sacrificial System
The whole system of animal sacrifice exercised by Abel, Noah, Abraham, and the other patriarchs finds its genesis in God's action here before it is codified in the Law of Moses.
The practice finds its ultimate fulfillment in the Suffering Servant of Isaiah:
He was despised, and forsaken of men, A man of pains, and acquainted with disease,
And as one from whom men hide their face:
He was despised, and we esteemed him not.
Surely our diseases he did bear, and our pains he carried;
Whereas we did esteem him stricken,
Smitten of God, and afflicted.
But he was wounded because of our transgressions,
He was crushed because of our iniquities:
The chastisement of our welfare was upon him,
And with his stripes we were healed.
All we like sheep did go astray,
We turned every one to his own way;
And the LORD hath made to light on him
The iniquity of us all.
He was oppressed, though he humbled himself
And opened not his mouth;
As a lamb that is led to the slaughter,
And as a sheep that before her shearers is dumb;
Yea, he opened not his mouth.
(Isa 53:3-7 JPS)
Animal Sacrifice for the covering of sin therefore prefigures the Messiah, who will accomplish this for his people. This is the seed of the woman whose stricken heal will crush the serpent's head.
Acted out by Abraham
We see an excellent example of this pattern in Abraham, the man of faith to whom the promises are given. Abraham appears to understand this principle and acts on it in faith.
He [God] said, “Take your son, your only son Isaac, whom you love, and
go to the land of Moriah, and offer him there as a burnt offering on
one of the mountains of which I shall tell you.” (Gen 22:2 ESV)
Though he is told to sacrifice his only son, whom he loves, he believes God will provide his own lamb and acts on this belief.
And Isaac said to his father Abraham, “My father!” And he said, “Here
I am, my son.” He said, “Behold, the fire and the wood, but where is
the lamb for a burnt offering?” Abraham said, “God will provide for
himself the lamb for a burnt offering, my son.” So they went both of
them together. (Gen 22:7,8 ESV)
And following through, he is stopped by God who does indeed provide his own sacrifice:
He said, “Do not lay your hand on the boy or do anything to him, for
now I know that you fear God, seeing you have not withheld your son,
your only son, from me.” And Abraham lifted up his eyes and looked,
and behold, behind him was a ram, caught in a thicket by his horns.
And Abraham went and took the ram and offered it up as a burnt
offering instead of his son. So Abraham called the name of that place,
“The LORD will provide”; as it is said to this day, “On the mount of
the LORD it shall be provided.” (Gen 22:12-14 ESV)
Intrinsic to the Meaning of Atonement
God's action of clothing Adam and Even is intrinsic to the meaning of atonement rather than incidental or secondary. Atonement is the very process of God providing a covering for sin through a blood sacrifice as he did literally for Adam and Eve.