Why do women wear veils over their heads during the Divine Liturgy?
Women wear veils to cover their hair during the Divine Liturgy in observance and obedience to the traditions delivered to them: "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. Every man praying and prophesying, having his head covered, dishonors his head. But every woman who prays or prophesies with her head uncovered dishonors her head, for that is one and the same as if her head were shaved. For if a woman is not covered, let her also be shorn. But if it is shameful for a woman to be shorn or shaved, let her be covered. Judge among yourselves. Is it proper for a woman to pray to God with her head uncovered? Does not even nature itself teach you that if a man has long hair, it is a dishonor to him? But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering" (1 Corinthians 11:1-15).
The spiritual meaning behind it is to cover human glory in the presence of God.