Paul says the woman should be covered because of the angels. He leads off by giving the created order.
But I want you to know that the head of every man is Christ, the head of woman is man, and the head of Christ is God. (1 Cor. 11:3 NKJ)
So, he lays out the order of authority. God is at the top, then Christ, then man, then woman. To show this authority Paul says the woman should be covered. This shows she is acknowledging her position in this order. Then he says she should be covered because of the angels.
8 For man is not from woman, but woman from man.
9 Nor was man created for the woman, but woman for the man.
10 For this reason the woman ought to have a symbol of authority on her head, because of the angels. (1 Cor. 11:8-10 NKJ)
Why because of the angels? It's because God is using the church to make known His manifold wisdom to the angelic realm.
To me, who am less than the least of all the saints, this grace was given, that I should preach among the Gentiles the unsearchable riches of Christ,
9 and to make all see what is the fellowship1 of the mystery, which from the beginning of the ages has been hidden in God who created all things through Jesus Christ2;
10 to the intent that now the manifold wisdom of God might be made known by the church to the principalities and powers in the heavenly places,
11 according to the eternal purpose which He accomplished in Christ Jesus our Lord, (Eph. 3:8-11 NKJ)
God is using the church to make His wisdom known to principalities and powers in heavenly places. This is why Paul says the woman should be covered and show the authority that is over her. Remember, when Satan sinned a third of the angels joined him and left their ranks, their positions. They rejected the authority over them. The head covering is for women to show that they accept their rank or position, the authority over them.