Genesis 3:20-24 meaning

Adam named his wife Eve which means living. Literally, Adam called his wife's name life because she was to be the mother of all living souls (v 20). God made garments of skin and clothed them (v 21). This is seen by many as the first animal sacrifice. To this point humanity was not given animals to eat, only plants; the permission to eat animals does not occur until Genesis 9. Blood was shed by the death of another in order to offer the garments that God provided by His grace. Later, Israel would offer animal sacrifices in order to remedy their sin.

When Adam and Eve disobeyed God, they did gain knowledge of good and evil which led to a life of strife and pain. The tree of life in the garden had fruit that would grant Adam and Eve immortality, an immortality accompanied with a life of strife and pain.

So, mankind was exiled from the garden. They became outcasts from paradise to work the ground from which Adam came. God placed angelic beings to guard the entrance to the garden, making the way to the tree of life inaccessible to mankind (v 24). Humanity was exiled from the place where immortality is possible. 

God promised that in the day Adam and Eve sinned they would experience death. What death's have they experienced so far? Death is a separation. Physical death is when the human spirit separates from the human body. Adam and Eve are now separated from paradise. They are separated from immortality. They are separated from a special relationship with God. They are separated from innocence. They are separated from creation, in that death had to take place to cover them. And they were even separated from themselves, seeking to cast blame on others rather than taking responsibility for their actions. Finally, by being exiled from the garden, they are condemned to physical death.

Chapter 3 tells the story of the fall of man into sin and out of his direct relationship with God. But, in it, we also see the promise of redemption through the Savior (Jesus).