Why do Muslim women wear the veil ?

The matter of women 's dress might seem trivial to some, especially in today's Western societies; however, Islam assigns to it moral, social and legal dimensions. Islam has defined the roles of men and women by allocating certain duties to each and granting right to each. This in order to maintain a proper balance in society. When men and women observe the proper Islamic dress, they not only protect their own honor and reputation, but they contribute greatly toward peace and order in society.
In general, there are certain guidelines concerning Muslim women's dress. Their garments should not be tight or translucent as to reveal the shape of what is covered. They must cover their entire bodies except the hands and face. This mode of dress is called "jilbaab" which refers to a women's outer garment, with which she is entirely covered. Muslim women do not dress modestly in obedience to their fathers, brothers or husbands, but only in obedience to God's commandments.
Both men and women are expected to be chaste and modest and avoid any type of dress and conduct that may invite temptation. Both are instructed to look only at what is lawful for them to see and to guard their chastity. Allah direct men first and then women in the Al-Qur'an:

(Tell the believing men to lower their gaze and guard their chastity. That is purer for them. Indeed, Allah is acquainted with what they do. Allah tell believing women to lower their gaze and guard their chastity and not display their beauty except what appear thereof; and to wrap part of their head covers over their chests and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husband's sons, their brothers or their brothers sons, or their sister's sons) {24:30-1}

The additional requirement for women to conceal their adornment and natural beauty is due to their greater need for privacy and protection. Except in the company of close relatives, a women is required to cover her entire body with loose fitting garments with the exception of her face and hands.
The Al-Qur'an states why Allah has prescribed particular dress regulations for women:

(O Prophet, tell your wives and daughters and the women of the believers to draw over themselves their outer garments {when in public}. That is more suitable so they will be known (to be pious believing free women) and not be abused...) {33:59}