God punished Adam and Eve because of disobedience. They disobeyed His word and the consequence of that disobedience caused separation from God and spiritual death. In Genesis 1 God gave Adam and Eve dominion over the earth.
"Then God blessed them, and God said to them, "Be fruitful and multiply; fill the earth and subdue it; have dominion over the fish of the sea, over the birds of the air, and over every living thing that moves on the earth." Gen 1:28
God in the beginning created everything perfect and everything was good, He walked daily with Adam and there was a communion and fellowship in the garden. God in Genesis 2 told Adam: "And the LORD God commanded the man, saying, "Of every tree of the garden you may freely eat; "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." Gen 2:16, 17
God said this to Adam before Eve was created and in Genesis 3 we read how Satan deceived Eve by twisting and questioning the word of God:
"Now the serpent (Satan) was more cunning than any beast of the field which the LORD God had made. And he said to the woman, "Has God indeed said, 'You shall not eat of every tree of the garden'?" And the woman said to the serpent, "We may eat the fruit of the trees of the garden; "but of the fruit of the tree which is in the midst of the garden, God has said, 'You shall not eat it, nor shall you touch it, lest you die. Then the serpent said to the woman, "You will not surely die. "For God knows that in the day you eat of it your eyes will be opened, and you will be like God, knowing good and evil." Gen 3:1-5
Satan questioned and twisted the words of God given to Adam and he lied to Eve by saying that they will surely not die. After Eve gave of the fruit to Adam, they did not die physically but they died spiritually. Though this one act of disobedience sin entered the world and sin brought forth death i.e. spiritual death, a spiritual separation from God. Adam and Eve tried to cover their sin by clothing themselves with fig leaves, which God did not accept, He clothed them with animal skins which signified that God instituted animal sacrifices and that blood was required to atone for sin. Adam and Eve fell for the lie and believed Satan rather than God and thus they forfeited the dominion over the earth to Satan. The Bible calls Satan the ruler of this world. "Now is the judgment of this world; now the ruler of this world will be cast out." John 12:31 Because of their disobedience, Adam and Eve were cast out of the garden, and had to labor and experience hardship, and later in Genesis we read about the results of sin i.e. Cain killing Abel and mankind became separated and increasingly wicked, which ultimately resulted in the judgment of mankind by God causing the earth to be flooded with water.
The disobedience of Adam and Eve also illustrates that God created man with a free will to choose, He did not created them as robots, He wants a genuine relationship with man that comes from the heart and not a artificial mechanical relationship that has no meaning. God looks at the heart of man and not on the outside appearances as mankind does. God declares in Jeremiah: "The heart is deceitful above all things, And desperately wicked; Who can know it?" Jer 17:9 Sin has caused mankind to be separated from God and God required blood to atone for their sins i.e. animal sacrifices. The blood of certain animals was the only way to cover the sins of the people in OT times. "For the life 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." Lev 17:11
Throughout the OT times people had to sacrifice animals until God gave His only son to shed His blood for mankind once and for all. Jesus was the perfect and unblemished sacrifice that was able to take away the sin of the world for all of those who would believe in Him through faith. Mankind cannot cover their sins on their own merit, we must come through Jesus who the only way to be reconciled to God the Father. "For as by one man's disobedience (Adam) many were made sinners, so also by one Man's obedience (Jesus) many will be made righteous." Rom 5:19
God is a just and righteous God and thus He cannot let sin go unpunished, light cannot dwell with darkness. "This is the message which we have heard from Him and declare to you, that God is light and in Him is no darkness at all." Jhn 1:5
God provided a sacrifice as He had promised to Abraham when he was tested to offer up his only son Isaac on Mount Moriah which is the same place were Jesus 2000 years later would give His life for us on Calvary's cross.
"Then He said, "Take now 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
"But Isaac spoke to Abraham his father and said, "My father!" And he said, "Here I am, my son." Then he said, "Look, the fire and the wood, but where is the lamb for a burnt offering?" And Abraham said, "My son, God will provide for Himself the lamb for a burnt offering." So the two of them went together." Gen 22: 7, 8
"And Abraham called the name of the place, The-LORD-Will-Provide; as it is said to this day, "In the Mount of the LORD it shall be provided." Gen 22:14
This promise God kept and fulfilled in giving His only son for us on the same mouton that Abraham was asked to offer Isaac. Sin has separated us, but we can be reconciled, forgiven and be born again spiritually through our Lord and Savior Jesus Christ who gave Himself for us.
"For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life." Jhn 3:16