This is like saying grace before meals…
Just as we say grace before meals, now the priest prays the prayer over the gifts.
There are three prayers which are unique for each daily Mass: the Opening Prayer, the Prayer over the Gifts, and the Prayer after Communion. If the Mass is in honor of the Blessed Mother, those three prayers will be related to her. If the Mass is in honor of a martyr, the three prayers will be related to that martyr, etc.
When the priest prays the Opening Prayer and the Prayer after Communion he invites everyone present to pray by saying, “Let us pray.” But he does not make this invitation for the Prayer over the Gifts, because the people just stood to pray the “Orate Fratres” and they are already in a disposition of prayer.
The rubrics state: “With hands extended, the priest sings or prays the prayers over the gifts, at the end of which the people respond ‘Amen.'”
Why does the priest extend his hands? Once again it is ‘body language’ that shows that he and all of the congregation are ‘open’ to the gifts that God wants to give them.