Well, let's take a minute and discuss first the facts, then the myth, and then the factors that went into that myth.
Fact: No one knows. Genghis Khan was an incredibly successful Mongolian conqueror whose empire was one of the most vast every recorded. It is highly probable that there are many people across the Asian continent who are distantly related, though through what circumstances we cannot know for sure.
Myth: According to popular legend, Genghis Khan commanded that every night, he have a new bed partner. This woman had to be beautiful, a virgin, and an inhabitant of his most recently conquered lands. His request was in part fueled by a desire to mix the bloodlines of the conquerers and the conquered, thereby creating one empire and one people.
Behind the Myth: Before genetics were well understood, people had to establish reasons for things that they found difficult to explain. You may heard the term "mongoloid" before; this is a now-offensive term for a person with Down's Syndrome. In an earlier era, Europeans found it impossible to understand why their children might be born with what appeared to them to be Asiatic features: thinner, almond-shaped eyes and a flatter, broader facial bone structure. In the racism of the same time, Europeans also saw themselves as by definition superior to everyone else, a belief that fueled many horrible atrocities. They decided that children with Down's Syndrome must have "inferior" blood in them, and used the legend of Genghis Khan's empire to support this belief. As a result, the raping myth was born.