There are many "interpretations" (which are not interpretations at all, but excuses) mostly for those who do not wish to recognize that a female is to have something other than longer hair down upon her head, to diminish her natural beauty when praying or telling others of God and His Son in situations acceptable to The Holy Ghost.
When thinking this out, consider 1 Corinthians 11:6 --
"For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered."
Now, it doesn't make much sense that if her hair is her cover, that it should be cropped close or shaved off, eh? No, the veil or mantilla or serviette, or something placed over her tresses to hide them, cannot be her hair. It must be something else. (Though one suspects that it should not be something wrought with a beauty to equal or surpass the glory of her own hair.)
Thayer's lexicon gives the underlying verb "to cover" (Strong's #2572) in the Greek of verses 5, 6, 7, and 13 as follows:
καλύπτω
kaluptō
Thayer Definition:
1) to hide, veil
1a) to hinder the knowledge of a thing
Part of Speech: verb
When uncovered or covered is spoken of here, think of a United States Marine, who never refers his(her) cap as a "hat"--it is always referred to as a "cover."
Why?
It is because when in uniform, the "cover" denotes his submission to, and an agent of, his authority--the "power" (Gk = ἐξουσία)issuing his orders and to which he owes explicit, minutely observed obedience.
Thus a Christian female obedient to Scripture and deliberately placing a cover down upon her head (and over its hair) shows not only her submission to The Godhead and His subordinately designated males, but also to the angels (both good and bad, including Satan) thus warning them of the mighty power The Godhead will confer upon her in praying and prophesying where it is her calling to do so.
Unfortunately, a rather inauspicious translation of verse 15 confuses the issue for the unwary or ignorant, by giving the English word "covering" to describe a completely different function of the female's hair:
"But if a woman have long hair, it is a glory to her: for her hair is given her for a covering."
The first century Koine-speaker would not have been confused at all, as you are, because the Greek word here is different, not connected with the theological sense of identification. It is Strong's #4018, for which the word "vesture" or "mantle" is the correct interpretation:
περιβόλαιον
peribolaion
per-ib-ol'-ah-yon
Neuter of a presumed derivative of G4016; something thrown around one, that is, a mantle, veil: - covering, vesture.
That is, a female's hair is a thing of beauty, and something to keep her neck and shoulders protected from heat, cold, and bruising. Periballeho literally means 'something thrown about one," like a wrap, or a scarf.
Now, the feminists do not like the idea that a woman should be expected, merely by the "accident" of birth as a female, to be required to be under the authority of designated males, nor to accede to it by an external symbolism like a bonnet or mantilla. But that is God's plan for the conduct of His churches in assembly, for the government of marriage and the family, and for representing Him as a gospel-bearing agent and nurturer to both unbelievers as well as demons.
Perhaps you might not be old enough to recall that throughout the ages, up until, say, the late sixties, women in any church service would usually be found covered. Does this explanation now make that passage a little more understandable to you?
With respect --
I am old enough to remember when hats were a requirement in the Catholic church although few women still wore hats to our Protestant church.