Anyone who's been around the left for a while will recognise the leading explanations for this anxiety. The first is the abiding legacy of orientalism and imperialism on Western cultures: the idea that the peoples of the East, specifically the Middle East to the Far East are culturally, politically, and technologically inferior to the West, and yet pose some kind of existential threat to it. Lump that in with the experience of imperialism as the colonising countries, you're left with a lingering set of racist attitudes and prejudices. True, in Britain overt racism has declined and anti-racism is officially promoted, but those roots have the habit of growing back and taking on new shapes. For example, "liberal" opposition to the niqab and Islam generally assumes women who wear the veil are victims forced to do so, or are unwitting dupes who collude in their oppression. This is because Islam is uniquely reactionary, etc. A variant of the liberal concern for Muslim women helped justify Bush jr's War on Terror, and fed the simplistic and intellectually bunk clash of civilisations thesis.
Orientalist, racist, and bastardised enlightenment ideas allied to geopolitical objectives is but one factor. Another oft-cited explanation is the pervasive quality of the 'male gaze', the taken-for-granted assumption that women's bodies can be sexually evaluated according to various sets of arbitrary but hegemonic criteria (smooth skin, pretty face, even proportions, ample boobs, shapely behind, fuckability), and that we all have a natural right to be able to so scrutinise them. Whole industries are based around the sexist but everyday cultural assumption that women are always on display. The niqab, which hides not just one's facial features but body shape challenges this "right" head on, even if ultimately its theological justifications are usually premised on equally patriarchal ideas. i.e. Only the husband has the right to gaze upon his wife's features. The assumption here is women are property, and need to be hidden away from men in general because they're beasts incapable of controlling themselves. When you have a culture likes ours in which the sexual objectification of women is mundane, you can understand why conservative ideas like these might find traction.
However, while I accept the validity of both these sets of ideas, that racism and sexism has a lot to do with opposition to the niqab, they fall into what you might call 'radical naivete'. This is something that overplays a given set of values, ideas and practices while, simultaneously, rendering a social problem a relatively simple matter to resolve. In this case if we keep steady progress uncovering and draining the reservoirs sustaining racism and sexism generally, the implication is people's attitudes toward niqab wearers will change and become more accepting almost automatically. I want to argue that the problems niqab wearers face, the general antipathy toward it is more complex. It taps into a series of social relations with long histories that combine with but are not reducible to the problematics of gender and ethnicity or, for that matter, questions of power differentials.
Jacques Derrida wrote books about the 'metaphysics of presence', the unstated assumption in Western cultures that the act of truth-telling is something that should be delivered in person, that the truth be enunciated verbally, that only presence can convey authenticity. The job interview, court case testimony, the confession, the lecture, these are instances and workings out of the privileging of presence. Unsurprisingly, bound up with this is an ethics and aesthetics of the face. You don't need to look into someone's eyes to see their soul, its secrets are given away by the sculpting and contorting of foreheads and lips. It is a banal observation, but the emphasis Western cultures place on face-to-face interaction means non-verbal cues - smiles, twitches, dancing eyes, frowns, the configuring of the face into expressions appropriate to the circumstances - are incredibly important to it. Anything that gets in the way of presenting the self via the face is jarring and sits very uneasily with it.
The second point flows from the first. The masking of faces have definite cultural connotations. There's no point stepping around the issue, in Western cultures from at least mediaeval times the obscuring of the face has been associated with villainy, criminality, and death. Think about our history - the masks plague doctors wore during the Black Death, the closed helms of knights sacking peasant villages for not delivering rent, the cloth covering the hangman's face, the depictions of highway men, artist impressions of Jack the Ripper, oxygen masks doled out during the two world wars, balaclava-wearing IRA and loyalist paramilitaries . The hidden face is unsettling and associated with those dark moments we'd rather not recall. Think about their modern iterations in popular culture. Villains from Scooby Doo to Darth Vader, from Michael Myers to Papa Lazarou, their masks and disguises convey upon them an aura of evil. In everyday life, the concealing of the face is associated with crime - typically robbery. See how the press and politics got themselves into a lather about hoodies, the strong association of them with anti-social behaviour and youthful thuggery. In this cultural context, quite apart from other factors, these deeply-held linkages mean the niqab wearer is from the outset is treated as a suspicious figure, as someone with something to hide and potentially up to no good.
In short, cultural anxiety toward the niqab is more than just sexism and racism. It taps into basic structures of feeling that are core to Western cultures. Does that excuse the antipathy and harassment niqab-wearing women face? No. It does, however, mean that tackling the issue is more complex than anti-racist and feminist explanations suggest.