Why does Allah command women to cover themselves?

Question

What exactly does it achieve? If a woman does not cover herself, how does that make her any less religious or less human? Some Muslims may say that it is degrading for a woman to show her body. Why should Allah care about a person's personal dignity as long as he/she is worshiping Him?

Answer 1

The verse that prescribes hijab for women is as follows:

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 khimār over their bosoms and not display their beauty except to their husband, 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.

Surat An-Nūr - ayah 31

Why does Allah command women to cover themselves?

Because that's good. Allah said in the Qur'an:

"Those who follow the messenger, the unlettered Prophet, whom they find mentioned in their own (scriptures),- in the law and the Gospel;- for he commands them what is just and forbids them what is evil; he allows them as lawful what is good (and pure) and prohibits them from what is bad (and impure); He releases them from their heavy burdens and from the yokes that are upon them. So it is those who believe in him, honour him, help him, and follow the light which is sent down with him,- it is they who will prosper."

corpus.quran.com/translation.jsp?chapter=7&verse=157

You might ask, why is it good?

Cause 1. Women who wear provocative clothes are more likely to get harassed. This scientific journal states:

The factual connection between revealing clothing and actual occurrences of sexual invasions is examined as well. The results demonstrate a gender-based attribution gap wherein men report perceiving the sexualized look as indicating an interest in sex and intent to seduce, whereas women cite their wish to feel and look attractive as its primary cause, while entirely rejecting the seduction claim.

Evidence attests to a fairly wide acceptance of these prejudiced collective beliefs among the general population, with over 50% of the public endorsing them to some degree or another (Buddie & Miller, 2002; Burt 1980). Significantly, men appear to endorse them to a considerably greater degree than women (De Judicibus & McCabe, 2001; Jimenez & Abreu, 2003; Kopper, 1996; Russell, 2004). Among the most prevalent of these allegations is a charge linking sexual aggression to the victim's so called "provocative", revealing wear ...

Cause 2. A women who doesn't cover will distract other Muslim man. If women don't wear decent clothes, and interact freely with man, it may lead to premarital sexual relations. This study concludes:

Conducted a laboratory experiment in which a male and female participated in a 5-min conversation while a hidden male and female observed this interaction. 36 sessions, involving 144 undergraduates, were completed. Male actors and observers rated the female actor as being more promiscuous and seductive than did female actors and observers. Males were also more sexually attracted to the opposite-sex actor than were females, and rated the male actor in a more sexualized fashion. Results suggest that males are more likely than females to perceive interactions in sexual terms and to make sexual judgments. Males seem to perceive friendliness from females as seduction, but this appears to be merely one manifestation of a broader male sexual orientation. (25 ref) (PsycINFO Database Record (c) 2016 APA, all rights reserved)

If a woman does not cover herself, how does that make her any less religious or less human?

It doesn't make her a less human but it does make her less religious because not covering is a minor sin. If she prays properly and follows all the principles of Islam, than her good deeds will erase her sins she committed by not covering.

Some Muslims may say that it is degrading for a woman to show her body. Why should Allah care about a person's personal dignity as long as he/she is worshiping Him?

Allah doesn't care if she covers or not. Allah prescribes covering because that's better for her. If she doesn't understand that and doesn't cover than she will commit a minor sin. People may blame her for it but they can't punish her cause there is no punishment prescribed for it.