In a classic Old West showdown, one gunslinger would typically outdraw the other, and he would be the winner. In other words, he would pull his gun from its holster faster and live to tell the story.
People sometimes also use the verb outdraw to mean "attract a larger crowd" — like the World Series might outdraw your favorite prime-time game show. The word draw has been used since around 1200 to mean "pull a gun out," and its Old English root dragan means "to drag or carry." And somewhere along the way, draw started to mean "attraction" — like the goody bags at the end of a birthday party are a big draw.