Here's a quote I've seen before. It really resonates:
When a woman veils her head she is shielding her heart to be wooed by the love of God in the Blessed Sacrament. This is a mystical ‘country’ that only the Eternal Father may enter. Her veil is like the lighted lamps of the virgins waiting for the Bridegroom, an indication that she is prepared to receive Him at a moment’s notice; an aureole of her spiritual love for the Bridegroom. Wearing the veil is an act of love of God.
It really is an outward sign: both for the wearer and for an observer. It isn't a way of attracting attention - far from it - but it does make me stop and think each time I enter a church. I can't just march straight in; I have to pause on the threshold of God's house while I put on the veil (it doesn't take long - I don't use a mirror or hairpins - just a few seconds to check it's the right way round before it covers my head) and those few seconds are a time of reflection: I enter into the presence of God and leave the world behind.
And then, in church, I can't rush around - I have to move carefully and gently, or the "flying nun" effect creeps in (or, given the absence of hairpins, my veil flies off!!)
Since I started to wear the mantilla, I have only forgotten it twice: once I got to the church porch, and felt that I was missing something, so went back to collect it from the car. The second time I was visiting a priest friend and hadn't expected to go straight into the church from the presbytery (the mantilla was again in the car, but going out to get it would have been awkward) - I felt intensely uncomfortable, almost "undressed," even though I was just being shown around the church.
Mhari says that she feels single women should wear white veils rather than black... for me, I think black is better: for a start, I nearly always wear black clothes, so sticking a white veil on top would really draw attention to me... but really, the important thing is to feel comfortable. If you are toying with the idea of trying a mantilla, I would heartily recommend it.