First let's consider the first commandment: Have no other Gods before Him. This should immediately bring up thoughts of the creation narrative. In Genesis 1 we don't see any interaction with other gods, we only see the assumed nature that there is only one God. We also see that we aren't to have any idols for number 2. In Genesis 3 we see Eve and Adam make an idol out of themselves by trying to become gods themselves through the forbidden fruit. Paul tells us that greed amounts to idolatry, that certainly would be true of Adam and Eve.
Next we seek that we shouldn't take the Lord's name in vain. Hopefully this brings to notion our relationship with God. We should know who God is, but also know Him personally. You cannot take His name in vain if you don't know Him. You might know the President of the United States, or you might just know who they are. This was broken when Adam & Eve ate the fruit, betraying God, whom they knew. The Sabbath should be plainly obvious being drawn from Genesis 1. Just as God rested we are to rest.
Honor our parents should be obviously from both Adam and Eve's rebellion against their Father in Heaven. The next few, do not murder, commit adultery, or bear false witness are also seen in this part of Genesis. Murder is brought about through the eating of the fruit, and thus death entered the world. Adultery shows purity which was lost when Adam and Eve realized they were naked and then bore false witness to God by hiding and then casting the blame one after the other from Adam to Eve to the Serpent.