Travel Without a Male Guardian

The Prophetﷺ said: 

"A woman must not travel alone without a mahram. No man may enter the house of a woman unless there is a mahram with her." 
A man stood up and asked Allah's Prophetﷺ: "O Messenger of Allah! My wife is going on Hajj ( pilgrimage), while I want to participate in a battle, what should I do?" The Prophet of Allahﷺ said: "Go out with her."
[Bukhari #1763]

For the protection of the person and honor of women, the Islamic rule encompassed in this tradition forbids any woman, whether she be young or old, single or married, to travel alone without (mahram) as a travel companion. This man must be of those to whom she is permanently prohibited to marry due to their close blood relationship, such as a father, a brother, an uncle, an elder son or nephew after he has reached puberty, or a husband, etc. Some people might say that this regulation restricts the freedom of the woman and her basic right of movement. The purpose of this rule is not to prevent her ravel but to free her from harm and molestation, and thus to maintain and preserve her right of dignity. 

Traveling entails many hardships and dangers, and since women are physically weaker than men, and moreover she may have circumstances of pregnancy, menses, nursing and childcare, or some ailments, she is in more need of help and service. Also women are generally more emotional and impressionable; they are more susceptible to unscrupulous and characterless men in search of victims. The Prophet of Allahﷺ expressed this most eloquently when he said to a man singing travel songs with a beautiful masculine voice in order to help the steady movement of the riding and pack animals, as per the custom of travelers:

"Go gently, O Anjashah, you're urging on the delicate glassware."

[Bukhari #5857] 

The phrase "delicate glassware" describes the fragileness and softness of the women's nature, who were the riders in that caravan, easily broken or disturbed. 
There are, as we all know, wicked minded, evil and violent men who lurk about and seek to take advantage of women whom they find susceptible, gullible or traveling alone. Such wicked-minded men are of course either interested in robbery, trickery, seduction, or rape. Therefore, a woman is very much in need of someone to care for, protect and fulfill her special needs while traveling, providing her the needed security, service and attention to help her in the difficulties and ward off strangers and potential predators. The "mahram" of a woman in Islam protects and serves with utmost sincerity since this is a natural obligation on him, rewarded by Allah (The Almighty). 

We find that many civilizations have similar rules about escorting women in travel, but often they allow other than the "mahram" to escort her since they do not differentiate between a "mahram" and a "non-mahram" in their cultures, which results in terrible stories. Therefore, and upon this line of reasoning, forbidding a woman to travel alone, and mandating a male escort who must be a "mahram", is not in any way a humiliating restriction or insult to her abilities, but, in fact, is an honor that she is served, protected and given companionship by a male escort who has to give her preference over his personal matters and needs.