Original answer below. First some notes on the accepted answer. It states three possible reasons:
1) Face coverings are perceived as an affront to feminism/women's liberation/etc...
This is fully compatible with the intro to my answer: "In a nutshell, the European reaction to everything they don't like is to try to ban it or to exterminate its practitioners" ... Europeans think Muslim women wearing the veil is an affront to feminism, so the first solution they come up with is to ban it. To the point of sending the police to the beach to force a woman to undress.
... There are worldwide "slut walks", whose main philosophy is feminism-based opposition
Seemingly missing the point that there is a difference between having the right to organize such protests versus dictating to others how they should dress.
2) It's viewed by Westerners that face coverings are symptomatic and/or symbolic of conservative/radical Islam's treatment of women.
So, the accepted answer agrees with mine that it is part of the de-Islamization drive ... similar to France's de-Christianization. They did not like Christian believers, so they chose to try to ban practices and exterminate practitioners.
3) For those who generally oppose radical Islam, this is viewed as both practical push-back (If you ban face coverings, conservative/radical Muslims wouldn't want to live in that society); as well as symbolic push-back (inasmuch as, per above point, face coverings are seen by many as a symbol of Islam).
And this is what I am warning against. First, the ban is not symbolic if people are fined for dressing a certain way, their children possibly taken away, or made to strip by police. 75 years of experience in Turkey shows even much stronger forms of repression does not dissuade the practitioners. Instead, such repression over trivial things creates becomes the rallying cry.
With that out of the way, here is my much down-voted answer:
This is an answer about the actual reasons behind these campaigns rather than the public excuses used to justify them.
Because there is a reason freedom-loving Europeans left Europe for America.
In a nutshell, the European reaction to everything they don't like is to try to ban it or to exterminate its practitioners:
During a two-year period known as the Reign of Terror, the episodes of anti-clericalism grew more violent than any in modern European history. The new revolutionary authorities suppressed the church; abolished the Catholic monarchy; nationalized church property; exiled 30,000 priests and killed hundreds more. ... New forms of moral religion emerged, including the deistic Cult of the Supreme Being and atheistic Cult of Reason, with the government briefly mandating observance of the former in April 1794. (emphases mine)
As Ben Cohen observes:
American Jews are fortunate to live with a constitution clearly demarcating religion and state. European Jews don’t have such clear guidelines, and therefore become hostages to the fortunes of political clashes in which their freedom of worship is just one consideration among many.
That's why French laïcité is oppressive whereas American separation of church and state (that is, no government at any level has the power to regulate people's religious observances) is liberating.
The freedom not to adhere to a specific set of religious rules is vacuous if one does not have the right to adhere to one's chosen set of religious rules:
The images of police confronting the woman in Nice on Tuesday show at least four police officers standing over a woman who was resting on the shore ... Her ticket, seen by French news agency AFP, read that she was not wearing “an outfit respecting good morals and secularism”.
European countries would do well to learn from the experience of Turkey which banned specific forms of dress for decades (and some people hanged for not complying with Atatürk's dress code):
REFİK BEY (Konya) — Bügünkü hakimiyeti milliye gazetesinin
neşrettiği giibi Sivas'da ve Erzurum'da şurada burada bazı masumlar, sırf (burada yapılan tahrik âmiz irticalkâr fikirlerin itesiriyle ızlâl edilerek ahkâmı kanuniyeye karşı hareket ederek müçrim vaziyetine düşmüşler, mahkûm olmuşlar, bunlardan bazıları idam bile edilmişlerdir. Nurettin Paşa da maksadını bu suretle istihsal etmiştir.
...
Ne günahı vardır Sivas'da asılan Necati'nin, ne günahı vardır Erzurum'da asılan bir kaç masumun? Doğrudan doğruya Nurettin Paşa tarafından verilen takrir memleketin dört köşesine yayılmış ve efendiler görüyorsunuz ya, bu kanun Teşkilâtı Esasiye Kanununa münafir, bu kanuni hürriyeti vicdana münafidir, (emphasis mine)
In this snippet, Konya representative Refik Bey expresses the view that the so called Hat Law violates the constitution of the Republic of Turkey and violates freedom of conscience during a debate on the law to ban certain religious institutions (Full disclosure: As far as I know, my great grandfather voted for both of these laws which I have considered for a long time to be, at the very least, misguided, to the consternation of many in my family).
That was Turkey's attempt to impose the French way and change people's behavior by coercion. In late 20th century, the headscarf ban became the rallying cry of political Islam against the establishment. The consequences are now here for all to see. Clearly, even very strict bans against religious expression did little to convert supporters of such religious expression over a period of 75 years. This establishes that such bans and repression do not improve domestic security, reduce conflict between different groups. Nor do they improve women's rights. Therefore, the real motivation behind such repression must lie elsewhere.
Current attempts at banning certain religious practices are not limited to just regulating dress, and are part and parcel of continental Europe's habit of ghettoizing people whom they consider to be beneath them.
As Yaakov Menken points out:
Given that kosher slaughter is in fact at least as humane as “stunning” via either electrocution or driving a peg into the base of an animal’s skull, to deny Jews and Muslims the ability to practice their religions, and claim it has nothing to do with bigotry, fools no one.
Update:
And, today, we have another gem:
“The still high incidence of punishable hate posting shows a need for police action,” Holger Münch, president of the Federal Criminal Police Office, said in a statement. “Our free society must not allow a climate of fear, threat, criminal violence and violence either on the street or on the internet.”
In the United States, despite the desire of many to institute Euro-style controls, hate speech is still protected speech.
