DNA origami is actually something of a misnomer; as William Shih notes, the process is more like DNA knitting. Think of a strand of DNA as a straight line--it has no area of its own. But by folding the line back and forth on itself, you can produce a flat plane.
In Rothemund's method, a length of single-stranded DNA folds back and forth to fill in a flat shape, be it a star, a square, or a smiley face. Then, shorter strands stick parallel and perpendicular to it, turning the single-stranded DNA into a double helix that links back to itself, stabilizing the shape. Finally, a computer tests the shape for structural stability. The finished product, to use the knitting metaphor, is a star, square, or smiley face-shaped blanket.