Most conspicuously, these old dragsters placed the driver behind the engine, and often behind the rear wheels. As the article explains, this decision was made for reasons of weight distribution and traction. In later decades, this decision was un-made for reasons of not having red-hot exploding motor pieces shooting into your face, as was often the case, among other motivations. See? Safety!
You may recognize Mickey Thompson's name as a record holder with his twin-engine car, mentioned toward the end of the article. The company he would later form still makes tires and wheels for all types of motorsports. Neat!
Anyway, there's also a 1959 documentary on the "exciting new sport of drag racing" at the bottom of the post, too - Ingenuity in Action. It's got exact music you think of, when dreaming of doing a pass in the high tens in your home built drag car: a sort of Holiday for Strings type of thing that was the soundtrack for everything in The Fifties, no matter how badass or potentially lethal the activity. It's frikkin' adorable. Don't miss it.