I'll give you what Adam Clarke says, and bold out the part I most agree with for my answer withing his answer.
Because of the angels - Some have explained this of good angels, who were supposed to be present in their assemblies (see Doddridge); others refer it to evil angels; and others to messengers or spies who, it has been supposed, were present in their public assemblies, and who would report greatly to the disadvantage of the Christian assemblies if the women were seen to be unveiled. I do not know what it means; and I regard it as one of the very few passages in the Bible whose meaning as yet is wholly inexplicable. The most natural interpretation seems to me to be this: “A woman in the public assemblies, and in speaking in the presence of people, should wear a veil - the usual symbol of modesty and subordination - because the angels of God are witnesses of your public worship Heb_1:13, and because they know and appreciate the propriety of subordination and order in public assemblies.”
According to this, it would mean that the simple reason would be that the angels were witnesses of their worship; and that they were the friends of propriety, due subordination, and order; and that they ought to observe these in all assemblies convened for the worship of God - I do not know that this sense has been proposed by any commentator; but it is one which strikes me as the most obvious and natural, and consistent with the context. The following remarks respecting the ladies of Persia may throw some light on this subject - “The head-dress of the women is simple; their hair is drawn behind the head, and divided into several tresses; the beauty of this head-dress consists in the thickness and length of these tresses, which should fall even down to the heels, in default of which, they lengthen them with tresses of silk. The ends of these tresses they decorate with pearls and jewels, or ornaments of gold or silver. The head is covered, “under” the veil or kerchief “(course chef),” only by the end of a small “bandeau,” shaped into a triangle; this “bandeau,” which is of various colors, is thin and light.
The “bandalette” is embroidered by the needle, or covered with jewelry, according to the quality of the wearer. This is, in, my opinion, the ancient “tiara,” or “diadem,” of the queens of Persia. Only married women wear it; and it is the mark by which it is known that they are under subjection “(oc’est la la marque a laquelle on reconnoit qu’ elles sont sous puissance o - power).” The girls have little “caps,” instead of this kerchief or tiara; they wear no veil at home, but let two tresses of their hair fall under their cheeks. The caps of girls of superior rank are tied with a row of pearls. Girls are not shut up in Persia till they attain the age of six or seven years; before that age they go out of the seraglio, sometimes with their father, so that they may then be seen. I have seen some wonderfully pretty girls. They show the neck and bosom; and more beautiful cannot be seen” - Chardin. “The wearing of a veil by a married woman was a token of her being under power. The Hebrew name of the veil signifies dependence. Great importance was attached to this part of the dress in the East. All the women of Persia are pleasantly apparelled. When they are abroad in the streets, all, both rich and poor, are covered with a great veil, or sheet of very fine white cloth, of which one half, like a forehead cloth, comes down to the eyes, and, going over the head, reaches down to the heels; and the other half muffles up the face below the eyes, and being fastened with a pin to the left side of the head, falls down to their very shoes, even covering their hands, with which they hold that cloth by the two sides, so that, except the eyes, they are covered all over with it. Within doors they have their faces and breasts uncovered; but the Armenian women in their houses have always one half of their faces covered with a cloth, that goes athwart their noses, and hangs over their chin and breasts, except the maids of that nation, who, within doors, cover only the chin until they are married” - Thevenot.
Does that work for you? lol