The question is a bit faulty. Let me clarify some points before answering it:
Muslim women can marry non-Muslims in non-Muslim countries if they want. However, such marriages are not recognized in Islam. Islam does not allow Muslim women to marry non-Muslims. The Quran says:
Do not marry unbelieving women (idolaters), until they believe: A slave woman who believes is better than an unbelieving woman, even though she allures you. Nor marry (your girls) to unbelievers until they believe: A man slave who believes is better than an unbeliever, even though he allures you. Unbelievers do (but) beckon you to the Fire. But Allah beckons by His Grace to the Garden (of bliss) and forgiveness, and makes His Signs clear to mankind: That they may celebrate His praise. - Quran 2:221
Muslim man are not allowed to marry all non-Muslim women. Only chaste women of people of the book are allowed. The Quran says:
This day are (all) things good and pure made lawful unto you. The food of the People of the Book is lawful unto you and yours is lawful unto them. (Lawful unto you in marriage) are (not only) chaste women who are believers, but chaste women among the People of the Book, revealed before your time,- when ye give them their due dowers, and desire chastity, not lewdness, nor secret intrigues if any one rejects faith, fruitless is his work, and in the Hereafter he will be in the ranks of those who have lost (all spiritual good). - Quran 5:5
It is important to note that lawful are not all the women of the People of the Book but only the chaste ones (chaste in this context means, the girls who desire chastity and refrain from illegal sexual intercourse). Both Muslim women and men are prohibited to marry polytheists, Buddhists and atheists.
Why are Muslim women prohibited from marrying non-Muslims?
The primary reason is because the Quran prohibits it. The Quran says in chapter 60 verse 10:
O ye who believe! When there come to you believing women refugees,
examine (and test) them: Allah knows best as to their Faith: if ye
ascertain that they are Believers, then send them not back to the
Unbelievers. They are not lawful (wives) for the Unbelievers, nor are
the (Unbelievers) lawful (husbands) for them. But pay the Unbelievers
what they have spent (on their dower), and there will be no blame on
you if ye marry them on payment of their dower to them. But hold not
to the guardianship of unbelieving women: ask for what ye have spent
on their dowers, and let the (Unbelievers) ask for what they have
spent (on the dowers of women who come over to you). Such is the
command of Allah. He judges (with justice) between you. And Allah is
Full of Knowledge and Wisdom. -- 60:10
If women were allowed to marry non-Muslim men, then her children would grow up in a environment where he is more likely to fall into disbelief. Naturally men are stronger and generally they have a dominant position in the house. Allowing a non-Muslim authority over a believer is not acceptable in Islam.