There are a number of reasons women choose to cover their heads during Mass. The most common is probably 1 Cor 11:3-15*, which I believe was the basis for requiring women to cover their heads pre-Vatican II. (But don't quote me on that.) Beyond that, though, there are a few reasons why I personally choose to do so: focus and humility.
Focus. I get distracted so easily: Oh, look, I know that altar boy... I saw his mom at the parish picnic last week... Picnics are fun - we should go on one more picnic before summer ends... Summer is almost over; we need to look for a snow shovel... I heard it snowed in Barrow the other night... Matt's going to Barrow for work soon... He hasn't been on a work trip in a month or so... Has the reimbursement check come in from his last trip yet?... The nice bank teller wants to meet the baby...
..And in just a moment or two, I've wandered so far down the rabbit trails that I have no idea what started it. Wearing something -- scarf, veil, hat -- on my head provides a tangible reminder of where I am, of what I need to be focusing on, of Whose presence I'm in. I can't count how many times my eyes have started to wander during Mass, been arrested by the bit of black lace at the edge of my field of vision, and snapped back up to focus on the sanctuary.
Humility. This was the reason I originally began wearing a veil: I realized that men are still expected, as per 1 Cor 11:4, to remove their hats in church. Why? To show respect. Then why don't women also follow St. Paul's instructions? Why is one instruction still standing after two millennia, and the other been discarded in the past 60 years? Why does the timing of women no longer covering their heads coincide so closely with the sexual revolution? Ah...
It seems plausible that the sudden cessation of veiling in church is linked to feminist claims that women should be treated exactly the same as men: men don't cover their heads in church, so women shouldn't have to. The problem is, if men remove their hats out of humility, to show respect, and women remove their headcoverings to prove they are as good as men, where is the women's humility before God? There are, I'm sure, many other ways of showing humility before God. For me, though, it has made more sense to follow St. Paul's recommendation and adopt the veil rather than find some other way.
Additionally, veiling helps me avoid being prideful about my hair. I have always dearly enjoyed dressing my hair... curls, braids, ribbons, rhinestones; I even went through a phase of weaving little bells into it. In high school, it wasn't uncommon for me to spend hours on a Saturday night putting my wet hair into 20+ French braids so that I could have a head of curly, bouncing locks for Mass the next day. Was I doing this for God, or to attract masculine attraction? Well, I *was* a teenager... Like Jo March, my hair was my "one beauty," and I intended to take full advantage of it.
Fortunately, it wasn't too much later that I realized the important difference between dressing well for Mass out of respect for God and dressing well for the sake of others' admiration. Today, wearing a veil during Mass ensures that I'm dressing up for the right reason, and is a way that I can offer my desire for admiration as a sacrifice to God.
Once again, I don't mean to imply that every Catholic woman ought to be covering her head at Mass -- that's not what the Church teaches. I have personally found the practice helpful, though, and I would encourage anyone who is considering it to give it a try!
* 1 Cor 11:3-15 (Douay-Rheims)
3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
4 Every man praying or prophesying with his head covered, disgraceth his head.
5 But every woman praying or prophesying with her head not covered, disgraceth her head: for it is all one as if she were shaven.
6 For if a woman be not covered, let her be shorn. But if it be a shame to a woman to be shorn or made bald, let her cover her head.
7 The man indeed ought not to cover his head, because he is the image and glory of God; but the woman is the glory of the man.
8 For the man is not of the woman, but the woman of the man.
9 For the man was not created for the woman, but the woman for the man.
10 Therefore ought the woman to have a power over her head, because of the angels.
11 But yet neither is the man without the woman, nor the woman without the man, in the Lord.
12 For as the woman is of the man, so also is the man by the woman: but all things of God.
13 You yourselves judge: doth it become a woman, to pray unto God uncovered?
14 Doth not even nature itself teach you, that a man indeed, if he nourish his hair, it is a shame unto him?
15 But if a woman nourish her hair, it is a glory to her; for her hair is given to her for a covering.