God had said in the day you eat of [the fruit of the Tree of the Knowledge of Good and Evil], you shall surely die. They ate and didn’t die, not in that day. Their eyes were opened, they became like God, knowing good and evil. All this is even confirmed by God. It seems the serpent told no lie at all. What happened?
Part of it may be their exile, which is a kind of picture of death. Perhaps in the same way that a branch cut from a tree does not immediately look dead, but being cut off from its source of life, it really no longer lives, so Adam and Eve, being cut off from the Tree of Life no longer really lived. Maybe. But they never ate from that tree. It wasn’t their source of life. God breathed into the man’s nostrils the breath of life and he became a living creature. Still, they were cut off from the sanctuary, from the presence of God.
A case can also be made out of the Hebrew grammatical construction. A stricter translation would read something more like in the day you eat of it, dying you shall die. Eating the fruit begins the process of ageing, decay, and death. The rest of their days would be a slow decline toward death. Dying, the process, you shall die, the end. This is certainly what we experience, and as far as that goes is good and true. I’m not sure the Hebrew is meant this way, though. This kind of construction is used throughout scripture for emphasis, as in the immediately preceding verse you may freely eat, or, eating you may eat—which is why nearly every English translation renders it you shall surely (or certainly) die.
A more complete answer might be found by digging into verse 21: And the LORD God made for Adam and for his wife garments of skins and clothed them. Where did the skins come from? An animal or animals had to die to provide the covering for them. This covering was given by God in place of the covering of fig leaves they had made for themselves. Why? To show that without the shedding of blood, there is no forgiveness—release, letting go, deliverance—from sin. This is the inauguration of the sacrificial system—or better, the provision of covering—later given more fully in Leviticus. The way back to the sanctuary is past the cherub and the flaming sword. Who can go through that? A man cannot enter, so he leans his hand on the head of an animal, identifying with it, sending it as his representative, and in a sense it carries him through the knife and the fire where it ascends to the glory cloud of the presence. This act, and most especially the blood spilled, provides a covering for the one who would draw near to God. This whole system is given by God as the way of approach. It is his provision, as is emphatically demonstrated by Jesus in his own death.
This helps (partly) explain why Abel’s offering was accepted and Cain’s wasn’t, and why Abel was a shepherd at all at a time when people didn’t eat meat (Gen 1:29-30, cf. Gen. 9:3). The fruit of the ground is a fine tribute or thanksgiving offering, but not an appropriate covering for the sin crouching at the door, trying to get mastery over him.