Short answer
They believe it pleases God (﷾), and they follow the wives of the Prophet Muhammad (ﷺ), who covered their faces. The niqab is a veil over the face that leaves the eyes open. The Qur'an tells believing women to draw their coverings over their chests (24:31). It tells them to put on their outer garments when they go out (33:59). Scholars differ on whether that includes the face. Most hold that a woman may show her face and hands. Imam Ahmad held that she covers them too (Dar al-Ifta). Women wore the niqab in the Prophet's (ﷺ) own time: he told women on pilgrimage not to wear it then (Bukhari 1838). In France, researchers spoke to 32 women who wore it. None had been forced, and most chose it themselves, often against their families' wishes (Open Society Foundations, 2011).
Full explanation
What the verses say
The Qur'an tells believing women "that they should draw their veils over their bosoms and not display their beauty" except to their close family (24:31). It tells the Prophet (ﷺ) to tell "thy wives and daughters, and the believing women, that they should cast their outer garments over their persons (when abroad): that is most convenient, that they should be known (as such) and not molested" (33:59). Aisha said what the women of Madinah did when 24:31 came down. They cut their waist sheets and covered themselves with the pieces. The common English translation reads that they covered their heads and faces (Bukhari 4759). The hijab covers the hair and neck and leaves the face open. The niqab adds a veil over the face that leaves the eyes open.
“And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women, or the slaves whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex; and that they should not strike their feet in order to draw attention to their hidden ornaments. And O ye Believers! turn ye all together towards Allah, that ye may attain Bliss.”
“O Prophet! Tell thy wives and daughters, and the believing women, that they should cast their outer garments over their persons (when abroad): that is most convenient, that they should be known (as such) and not molested. And Allah is Oft-Forgiving, Most Merciful.”
Safiyya bint Shayba reports that Aisha used to say that when the verse telling the women to draw their veils over their bosoms (24:31) was revealed, they cut their waist sheets at the edges and covered themselves with the pieces. The common English translation reads that they covered their heads and faces.
The wives of the Prophet (ﷺ)
For the Prophet's (ﷺ) wives the Qur'an set a further rule. Men who needed something from them were to ask "from before a screen: that makes for greater purity for your hearts and for theirs" (33:53). Aisha tells that once, when she was left behind on a journey, Safwan ibn al-Mu'attal found her. He knew her because he had seen her before the veil was ordained. She covered her face with her garment (Bukhari 4750). This is the example that women who wear the niqab follow.
“And when ye ask (his ladies) for anything ye want, ask them from before a screen: that makes for greater purity for your hearts and for theirs.”
Aisha's own account of the slander spread against her. She says the Prophet (ﷺ) went a whole month without receiving any revelation about her case, while the rumour spread and he consulted his Companions, before the verses declaring her innocence came down. When Safwan ibn al-Mu'attal found her left behind and knew her, for he had seen her before the veil was ordained, she says: I covered my face with my garment.
Required, or chosen? The schools
Egypt's official fatwa body sets out the schools of Islamic law. The majority of scholars, it says, hold that a woman covers all of herself before men outside her family except her face and hands, and it quotes scholars of the Hanafi, Maliki and Shafi'i schools for this. Imam Ahmad held that she covers all of herself, even her nails (Dar al-Ifta). So some women wear the niqab because they hold it is required, and others choose it as an extra act of devotion. Elderly women, the Qur'an says, may set aside their outer garments, "but it is best for them to be modest" (24:60). On what she herself should do, a woman asks a qualified scholar.
On whether a woman must cover her face. It says the majority of scholars hold that a woman's whole body is to be covered before men outside her family except her face and hands, and gives the schools: the Hanafi al-Marghinani, that her face and hands are excepted; the Maliki al-Baji, that a man may look at a woman's face and hands; the Shafi'i Ibn Hajar al-Haytami, quoting Qadi Iyad, that she does not have to cover her face, and that covering it is recommended, not required; and Imam Ahmad, that all of her is to be covered, even her nails, while the Qadi of the Hanbali school excepted the face and hands. It sides with the majority: she may show her face and hands.
“Such elderly women as are past the prospect of marriage,- there is no blame on them if they lay aside their (outer) garments, provided they make not a wanton display of their beauty: but it is best for them to be modest: and Allah is One Who sees and knows all things.”
Women wore it in the Prophet's (ﷺ) time
The Prophet (ﷺ) told the woman in ihram, the pilgrim's state, not to wear the niqab or gloves (Bukhari 1838). A rule against wearing it on pilgrimage shows that women were wearing it the rest of the time. The niqab did not come from a later age.
Ibn Umar reports that a man asked the Prophet (ﷺ) what may be worn in ihram, the pilgrim's state. He listed what men may not wear, and said that a woman in ihram should not wear the niqab, the face veil, or gloves.
Covering is older than Islam
Veiling did not begin with Islam. In the Torah, when Rebekah saw Isaac coming to meet her, "she took a vail, and covered herself" (Genesis 24:65). Paul told the women of Corinth to cover their heads when they prayed (1 Corinthians 11:5-6). Until 1983, Catholic church law told women to cover their heads in church (canon 1262 of the 1917 Code).
“For she had said unto the servant, What man is this that walketh in the field to meet us? And the servant had said, It is my master: therefore she took a vail, and covered herself.”
“But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. 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.”
Canon 1262, section 2, of the Catholic Church's 1917 Code of Canon Law directs that in church men attend with heads uncovered and women with heads covered and modestly dressed, above all when they come to receive communion. The 1983 Code, which replaced the 1917 Code entirely, does not repeat the rule.
Who chooses it, and why
In a study published in 2011, the Open Society Foundations spoke to 32 women who wore the full-face veil in France. Twenty-nine had been born in France, and eight were converts. None had been forced to wear it. Most had chosen it themselves, as part of a spiritual journey, and often against the wishes of their families, above all their mothers. Thirty had been verbally abused in public. All said they were willing to show their faces for identity checks (Open Society Foundations, 2011).
A study of 32 women who wore the full-face veil in France, published as the law banning it came into force; the French government said the ban was needed to protect equality between men and women and public order. Twenty-nine of the women were born in France, thirty were French citizens and eight were converts. None had been forced to wear it. Most chose it themselves as part of a spiritual journey, many taking the Prophet's (ﷺ) wives as their example, often against the wishes of their families, and above all of their mothers. Thirty had been verbally abused in public. All said they were willing to show their faces for identity checks.
The law where she lives
France banned clothing that hides the face in public in 2011. The same law made forcing anyone to hide their face a crime, punished with a year in prison (French law 2010-1192). A Muslim keeps the law of the country where she lives, for the Qur'an says: "fulfil (every) engagement, for (every) engagement will be enquired into" (17:34). Where the face veil is banned, or an official needs to see her face, she asks a qualified scholar what to do.
Article 1: no one may wear in public space clothing meant to hide the face. The law came into force in April 2011, with exceptions for health, work, sport and festivals. It also made forcing anyone to hide their face, by threat, violence or abuse of authority, a crime punished with a year in prison and a fine of 30,000 euros.
“Come not nigh to the orphan's property except to improve it, until he attains the age of full strength; and fulfil (every) engagement, for (every) engagement will be enquired into (on the Day of Reckoning).”
A study of 32 women who wore the full-face veil in France, published as the law banning it came into force; the French government said the ban was needed to protect equality between men and women and public order. Twenty-nine of the women were born in France, thirty were French citizens and eight were converts. None had been forced to wear it. Most chose it themselves as part of a spiritual journey, many taking the Prophet's (ﷺ) wives as their example, often against the wishes of their families, and above all of their mothers. Thirty had been verbally abused in public. All said they were willing to show their faces for identity checks.
Perspectives
The niqab oppresses women
The face veil is forced on women, hides who they are and is not even required by the Qur'an.
Women are forced to wear the niqab.
Rebuttal
Forcing it is wrong, and France made it a crime. But when researchers asked 32 women who wore it in France, none had been forced, and most had chosen it against their own families' wishes (Open Society Foundations).
In the West the niqab is controversial: it is said to stop women fitting in, and to be backward.
Rebuttal
Many who defend a woman's right to wear a bikini object when she chooses to cover herself. Both are her own choice, and forcing either on her is wrong. France made forcing the face veil a crime. The women in the French study were nearly all born in France and were French citizens. They chose the veil themselves, and were willing to show their faces for identity checks (Open Society Foundations). The answer is the same both ways: it is the woman's choice.
The niqab is not in the Qur'an.
Rebuttal
The word is not, but the Qur'an commands women to cover (24:31, 33:59). Scholars differ over whether that includes the face. Imam Ahmad held that it does (Dar al-Ifta). And women wore the niqab in the Prophet's (ﷺ) own time (Bukhari 1838).
A covered face hides a person's identity.
Rebuttal
Every woman in the French study said she was willing to show her face for identity checks (Open Society Foundations).
The Islamic perspective
The Qur'an commands women to cover. Scholars differ over whether that includes the face: most say a woman may show her face and hands, and Imam Ahmad held that she covers them too. Women who wear the niqab follow the wives of the Prophet (ﷺ).
The Qur'an tells believing women to cover, and to put on their outer garments when they go out.
Most scholars hold that a woman may show her face and hands; Imam Ahmad held that she covers them.
Women wore the niqab in the Prophet's (ﷺ) time, and his wife Aisha covered her face.