[FYI, I am a Ph.D. candidate studying Modern Middle East history.] An answer to this question requires some background on the history of veiling as an Islamic practice.

Many academics and historians, both Muslim and non-Muslim, have argued that full body veiling was not an intrinsically "Islamic" practice. Some scholars have argues that the pre-Islamic Arabs, whether male or female, usually wore cultural garb that included a head covering and long robes, perhaps as protection from the intense heat of the desert. Until today you will still see men in the Gulf wearing long robes. We cannot know with certainty what Arabian women wore before Islam; their cultural dress may or may not have included covering faces. Yet, this certainly became a practice after the arrival of Islam. Let's get into how this happened.

As Usama Hasan writes, "Of over six thousand ayat [verses] of the Quran, only about half a dozen refer specifically to the way a woman should dress and/or walk in public. Only two ayahs [verses] mention items of clothing, i.e. the khimar (headcover) and jilbab (outer garment) respectively." (Source: https://www.academia.edu/4386604/Islam_and_the_Veil_-_Usama_Hasan)

Thus, the Quran does not explicitly lay out what kind of clothing women should wear and what should be covered (aside from mentioning that women should draw a shawl over their bosom in Chapter 24: Verse 31). Instead, most of the rulings on women's dress came about because of another class of traditional Islamic sources: the "hadith," or sayings of the Prophet Muhammad. There are different hadith that state guidelines on what women should wear, but it is hard to prove their authenticity and many are contradictory and fabricated. The article I linked to above delves into various hadith on the subject.

After Muhammad allegedly received divine revelation and began spreading the religion of Islam in Arabia, his first wife died. He subsequently married multiple women, most likely to facilitate the creation of new alliances between tribes or establish some new legal rulings on marriage through example. Then a new verse of the Quran was revealed, stating "And when ye ask (the Prophet's ladies) for anything ye want, ask them from before a screen: that makes for greater purity for your hearts and for theirs" [Chapter 33: Verse 53]. This verse established the concept of segregation specifically for the wives of Muhammad. The verse commanded men who were not related to Muhammad's wives to not directly look at or speak to them without some kind of barrier. Scholars such as Fatima Mernissi and Leila Ahmed argue that this was the beginning of the practice of covering faces, though it began as an actual physical barrier, not just a piece of clothing. But when the Prophet's wives went out, which they did often (as is demonstrated by many hadith that mention the wives speaking to men or even negotiating during warfare, like Aisha in the Battle of the Camel), they would therefore veil their bodies completely.

Scholars argue that other Muslim women perhaps began to imitate the wives of Muhammad and similarly veil their bodies and faces. Please see Leila Ahmed, Women and Gender in Islam: Historical Roots of a Modern Debate and Fatima Mernissi, The Veil And The Male Elite: A Feminist Interpretation Of Women’s Rights In Islam. Of course, these feminist Muslim scholars' works are much more nuanced, but I'm just trying to summarize some of their arguments!

Once Muhammad died, the following generations began codifying Islamic law by compiling the Quranic verses into one canon volume and collecting different sayings of Muhammad (hadith). As Mernissi and Ahmed point out, most of the scholars who participated in this movement were men. Because they arose in a patriarchal society, they interpreted Islamic texts through that frame of reference and formulated laws that were often misogynistic and took away some of the rights that Muhammad supposedly sought to establish for women. Therefore, most early Muslim scholars argued that veiling and the segregation of the sexes is required in Islam.

In the centuries after Muhammad's death, Islam spread quickly. In addition, practices like full-body veiling that the early Arabian Muslim scholars endorsed also spread. Many people throughout the Middle East and North Africa embraced Islam for pragmatic reasons; adhering to its conventional practices provided an opportunity for social, political, and economic mobility in the new Muslim-ruled empires. In cultures where Islam became the dominant religion, even other religious minorities would adopt some of its practices. Thus, veiling in public became a cultural norm in many places. It was often more common in urban areas among upper-class women, because such families could afford to keep women segregated.

Fast forward many years to the late 19th and early 20th century. The Ottoman Empire, which controlled so much of the Middle East as well as parts of North Africa and Europe was in decline; European colonialism and imperialism was at its peak. For a variety of reasons (including exposure to the nascent Western feminist movement), a new homegrown Islamic feminism arose. Major thinkers such as Qasim Amin in Egypt argued that veiling was merely a cultural practice that was not rooted in true Islam. They believed that the face veil and the practice of secluding women in homes with no contact aside from male relatives was not only holding back women, but society as a whole. It prevented women from contributing to the new national movements that arose in the aftermath of colonialism. As new interpretations of Islam and new secular nationalist movements arose, many Muslims began to eschew the practice of veiling. Women increasingly began playing a greater role in society and did not wear face veils, though many continued to cover their hair for cultural or religious reasons. By the 1950s and 1960s, it became common to see Muslim women in different classes, whether from Lebanon to Pakistan, who didn't even wear a full headscarf, much less cover their face.

However, due to a host of reasons that is beyond the scope of this thread, a fundamentalist version of Islam became more popular throughout the Middle East, North Africa, and Central and South Asia. (Ok, to quickly summarize: By the 1970s and 1980s, many popular secular movements had been quashed and numerous post-colonial states were ruled by corrupt, tyrannical dictators. These leaders often espoused a secular nationalist ideology while practicing a tyrannical form of rule. Thus, people turned to the idea of an Islamic political system as a means of resistance. Also, the rise of Saudi Arabian oil money, and their willingness to donate to various causes under the condition that their Wahabbi ideology of Islam also be adopted, spread fundamentalist views throughout Muslim communities. This doesn't even get into the role that European and American interventions in the Third World played in sparking new Islamist views.)

In conclusion, veiling in the form of either a headscarf or even the face covering has resurged in popularity in recent decades, whether by force or by choice, because different regimes and communities have embraced a return to a traditionalist, fundamentalist version of Islam. New fashions in wearing a headscarf or a full body veil have also arisen. As more and more Muslims have emigrated from their homelands and have created new diaspora communities, many Muslim women choose to wear a headscarf to make a statement about their religiosity or identity. Some women even argue that it can be a feminist practice. For more on this, read the work of Amina Wadud or Leila Ahmed's 2011 book entitled Quiet Revolution: The Veil's Resurgence, from the Middle East to America.

TL:DR - After the rise of Islam, veils became common for different reasons, one of which was that the wives of the Prophet Muhammad adopted the practice and then other Muslim women began to emulate them. Furthermore, the early Islamic scholarly class was rooted in a misogynistic environment and interpreted various Quranic verses and Prophetic sayings to argue that full body veiling, or partial veiling (covering everything except the face, hands, and feet) was required. However, by the early 1920s, the rise of the Muslim feminist movement led many women and men to shun face veiling and often the veiling of hair. This huge revolution led to veiling becoming unpopular and way less common in major cities throughout the Middle East, North Africa, Central Asia, and South Asia by the mid-20th century. However, the recent rise of Islamism (a fundamentalist view of Islam that views the religion as a political doctrine) led to the resurgence of the veil as both a cultural and religious practice. More recently, some Muslim women have started to view wearing a headscarf as a choice and a feminist practice.