First of all, head covering is not only in Islam. Both Jewish and Christian traditions had veil for women in the history; such as Mary (Maryam) in the history, or current nuns, or Amish and Mennonite women who are still covering their head as a type of modesty. So it did not start with the religion of Islam.
Veil means obeying one of the commands of God in Islam. It means modesty. Believing men and women are supposed to lower their gaze and guard their modesty according to the Qur’an (24:30,31). The women should not display their beauty and ornaments to the opposite gender outside of the family.
The whole idea is preserving the modesty. Why is modesty important? Modesty is prescribed to protect women from molestation (Qur’an: 33:59). The purpose of the veil in Islam is protection women from evil intentions of the opposite gender. Woman’s covering her beauty and ornaments means hiding them from evil looks. Hair is considered as the beauties of the women as well.
Furthermore, covering the head is not enough to be modest in appearance. Men and women are not supposed to wear very tight and transparent clothes so that they don’t expose the curves of the body and by this way preserve their modesty. Men also have to cover certain parts of their body (from belly button to the under nee). The difference between the two genders in clothing is because the difference in the creations and natures of the two genders. It is not only modesty in cloth, but also modesty in speech and manners of both men and women are required in Islam.
It is better for a Muslim woman to think that ‘my body is very important like a jewelry in Allah’s sight, and I am preserving my body and hair just for the sake of Allah.’ It is not a sign of man’s authority over woman, but it is sign of woman’s submission to Allah only.
One may say that covering woman is more likely to protect themselves from sexual assault because they are already giving the impression that they want to protect their modesty in public and want to be safe against illicit sexual relations.
The whole idea of covering is not to excite the opposite side physically and not to be sexually desirable by the opposite gender. This is one of the ways of preserving the individuals in the society against unlawful sexual relations. Modesty is one of the ways for staying safe and not crossing the line.
Finally, according to the testimonies of the covering Muslim women, they feel more respected by men in general and more liberated from sexual looks of the opposite gender actually and feel more peaceful at heart when they covered their heads.