Now is this because God somehow needs to rest as we do? It seems contrary to the uniform description of God as the Almighty who does not grow tired (Isaiah 40, Psalm 121). Most of us have read this at a simple level. From the greater to the lesser: If God has to rest, so should you.
What if we apply the concept that John Walton suggests. In Ancient Near East culture, a “god” rested in its temple in the sense that it took possession and assumed control. So the rest of God is not that he is akimbo in a hammock. It is rather that on the 7th day (which is not terminated in Genesis 1 like days 1 through 6) God assumes possession and control of the heavens and the earth as his dwelling place.
We then rest, not in imitation, but to show that we trust that the Almighty can in fact take care of us on that one day in seven when we do not work, or the one year on 7 when the land rests or the one year in 49 when debts are forgiven. Our rest is not imitation but response.
This makes more sense theologically. It fits the core meaning of the word for sabbath, which means to stop or cease. God stopped creating (because he was done, and the world was “very good”); we cease from working to rest and to enjoy life with our Creator.