You can study what the Church Fathers and the Sacred Scriptures stated about wearing a veil as well as the tradition's profound theological and symbolic meanings in this lovely thirteen-page booklet. You'll learn what a woman's veil indicates about her:
Esteemed vocation in God’s creation
Intimate spousal relationship to Christ
Sacredness as a tabernacle of life
Mystery as a “garden enclosed”
Adoration for God and praise of His glory
Humility and receptivity to God’s love
Most importantly, you'll learn how donning the chapel veil contributes to the testimony of and renews faith in the Real Presence of Jesus in the Holy Communion. You'll learn how women's dignity and their priceless gift of self to the world are reflected in how women wear veils as well.
This is an excellent explanation of the theology of the chapel veil. It makes it easy to explain to others who are considering the veil, or just curious, why we veil in the Presence of the Blessed Sacrament.