Exile is a form of punishment that God has used from the very beginning. Here in Genesis 3, in the book of beginnings, we have the first instance of exile imposed by God Himself. It was exile from the Garden of Eden, from all that was wonderful and good that God had created, the perfect environment in which He had placed Adam and Eve. They could never go back. God placed an angel with a flaming sword that would turn whichever way any man juked to get back. If it were still there, it would deny us "paradise" even now.
This context shows three reasons we can glean to determine why God uses exile. The first one is evident—it was punishment for their sins. Adam and Eve took of the tree of the knowledge of good and evil when God said they should not take of it. That is sin, breaking a direct command of God. Exile was the punishment.
What else can we glean? What did their exile do? It separated them from access to Him. So, secondly, exile separates man from God. He does not want to be separated from us, but because of sin, it happens. It must happen because He does not like sin in the least. So this is a kind of corollary to the first point. Sin brings exile, and sin causes separation from God.
The third point must be read into it, but it is obvious from God's intent and the way God is. God imposes exile to spur repentance because it should be the natural inclination of men who have known God and all the glorious things that we can have in His presence to return to His good graces.
In summary, the first point is exile occurs because of sin. The second point is exile happens because sinners must be separated from God. And the third point is God uses exile as a goad to motivate sinners to repent.