The concept of the bishounen is actually quite old, predating anime, and even predating manga as we know it today.
The concept dates back to early kabuki and similar forms of Japanese theater. Traditionally, all of the parts in these plays, even female roles, were always played by men, and in some genres this tradition continues to hold true. There are several theories as to why this practice came to be, ranging from fears that actresses would be used for prostitution to the idea that men can play women's roles better than women can.
Whatever the cause of the tradition, however, it led to a group of actors known as onnagata, or men who specialized in playing female roles. Naturally, it was considered a Good Thing for such men to look feminine, as it enhanced the illusion. Very often, they carried their "feminine" ways away from the stage, dressing, speaking, and acting as women even in real life. Although there are fewer onnagata nowadays, as more genres of Japanese theater are being opened to female players, they can still be found. The specialization is highly respected, in a way which may seem foreign to many other cultures.
At one time, onnagata were considered the ideal of femininity in Japanese culture. Even geisha would go to them to pick up pointers. Perhaps this could be taken as yet more proof that the Japanese are truly mad. Or perhaps it explains why bishounen are so popular today, as a remnant of that ideal to which some women still aspire. Or maybe people just think they're kawaii.
As a final note, the term bishounen is only accurately applied to young and teenage boys. Grown men who would otherwise qualify, such as Hotohori and Zagato from Tilyrna's writeup are more correctly called biseinen. However, this term is not very popular among fans, and is not commonly heard anymore except in the occasional review