Why are Muslim men allowed to marry a non-Muslim but Muslim women are not allowed?
One theory is due to genetic and psychological differences between men and women, Muslim men will be better able to remain practicing Muslims and to bring up devout Muslim children even if their wives are Christian or Jewish, while Muslim women will be less likely to accomplish these.
Scientific studies are needed to prove whether the above is true, but it seems to be true from anecdotal evidence. It is probably true that some Muslim women will be perfectly capable of remaining practicing Muslims and bringing up practicing Muslim children when married to non-Muslims, but these will be the exceptions, not the rule. It is similar to drinking wine; some people are able to enjoy it without becoming drunkards, but Islam forbids it to all Muslims since this is better for everyone. So that fact that I or you can drink wine without it causing us noticeable harm does not mean that it is halal for us and haram for others. It is haram for everyone.
If Muslim women who are married to non-Muslims are twice as likely to stop practicing Islam compared to Muslim women married to Muslim husbands, and/or if their children are twice as likely to abandon Islam, then these can be considered sufficient justification for prohibiting it.
So in this case, as in the case of wine-drinking, a person has to refrain from it for the sake of the greater good.
There might be many other reasons for forbidding such marriages, I’m mentioning only two potential explanations.
In Islam, everything is allowed unless explicitly forbidden. In the matter of sex, however, the Quran reverses matters; everything is forbidden, as numerous verses say, unless explicitly allowed. The Quran commands the believers to “guard their privates” (abstain from sex) in five places (23:5, 24:30, 24:31, 33:35, 70:29), then in the contexts of two of these verses it makes exceptions for cases of lawful relationships (23:6, 70:30). The picture that the Quran draws is that all sexual activity is forbidden, except when it is expressly allowed. Since the Quran expressly allows men to marry non-Muslim women belonging to God’s other religions, while it does not expressly allow women to do its counterpart, this can be considered strong evidence for considering the latter forbidden.
Those who want to legalize marriage between Muslim women and non-Muslim men say that such marriages are in a gray area, even though they are not expressly allowed, they are not expressly forbidden either. The reasoning offered by such people is that it is in the best interests of Muslim women to be allowed to marry outside the faith, that this is more likely to ensure their long-term good, and that the prohibition may have made sense in certain societies, but does not make sense in Western-style diverse and multi-religious societies.
But as I mentioned above, if such women and their potential children are more likely to abandon Islam, then that is a very good reason for prohibiting such marriages, and it is a weak argument to say that it is materially better for women to marry outside the faith if their spiritual and eternal life is harmed by this.
Islam is not forced on people, so a Muslim woman should be free to marry outside the faith from a civic law perspective, so it is ultimately a matter between the woman and God. Marriage is one of the most important decisions in life, and a Muslim woman who truly fears God and wishes to please Him will never base her marriage on what is at best in a gray area, since she cannot be sure if God will be pleased with her.