God the Son was a Person, a Someone, possessing the nature of God in its fullness, and this in the eternity of the Divine Being. At a certain point in time He took to Himself and made His own a human nature. Thus we have the unique instance of one single person with two natures, divine and human. To the question “Who are you”? Christ would have but one answer. He is the Second Person of the Blessed Trinity, God the Son, the Word. But to the question “What are you”? Christ Our Lord would have two answers, for He has two natures; He is God and He is man.
What Sheed explains is that the person and the natures must be understood when examining the mystery of the incarnation. It was the person of the Son who became incarnate. But that person possessed two complete natures: God and man.
How this can be is simply a mystery. But this does help us understand who and what we are talking about when we think of the incarnation.