All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
A Muslim can marry the chaste women from the People of the Book. But a Muslim woman cannot marry any non-Muslim man. The main reason for this ruling is that Allah Who is The Supreme Legislator permitted Muslim men to marry chaste women from the People of the Book and prohibited Muslim women from marrying a non-Muslim, be he a Jew, a Christian, or any other disbeliever.
No one has any right to object to the Ruling of Allah, the Lord of all creatures.
A Muslim believes in all Messengers and Prophets of Allah. He cannot be a Muslim unless he believes in all Messengers of Allah; among whom are Prophets Moses and Jesus, may Allah exalt their mention. Thus it is permissible for him to marry a Jewish or Christian chaste woman since he respects her right to her religion and believes in her Prophet and he also knows that if he degrades them by any way or any means, he is no longer considered a believing Muslim.
Moreover, a Muslim husband is commanded to be kind to his wife from the People of the Book. He also has to treat her with good manners and not force her to embrace Islam, and she is entitled to the same rights that are due to a Muslim wife. From the above mentioned facts, a Muslim man is allowed to marry a Jewish or Christian chaste woman while a Muslim woman is not allowed to marry a non-Muslim man.
Actually, Allah honored the Jewish or Christian women when He made them appropriate wives for Muslim men though these women do not believe in Islam, or in the Prophet Muhammed . For the same reason Allah has forbidden a Muslim woman to marry a non-Muslim man as the latter neither believes in her (Muslim woman's) religion nor in her Prophet. Women usually follow their husband's footsteps and generally a husband has an upper hand on his wife. Moreover the husband most of the time is the one who takes care of the wife and provides for her needs. So how could it be possible to permit a Muslim woman to be under a non-Muslim man who denies her religion and makes jokes of her Prophet! So, only when the man believes in Islam and in the Prophet Muhammad can he then marry a Muslim woman.
In fact, Judaism and Christianity ordered their followers to believe in the Prophet Muhammad who will come after their Prophets. Though the Christians and the Jews have made many changes to their Books, there are still glad tidings of the coming of the Prophet Muhammad in their Books.
Allah knows best.