Try the Paper Folding Construction of an Ellipse first. Compare that construction method with this one. You will see that the constructions are identical. The only difference is whether the second focal point, F2, is moved inside or outside the circle. Trace the hyperbola with X, after selecting one of the tracing options. You can vary the dimensions of the hyperbola by moving the other red control ponts.
To do this construction with paper start with a printout of a circle with many radial lines drawn. Mark a point outside the circle. Cut out a little hole at that point. Fold the hole to each point on the circle and make a crease. The envelope of creases will trace out the hyperbola. The actual point on the curve is where the crease intersects the radial line to the point.

Proof that this constructs a hyperbola
Note that the distance PX always equals the distance PF2. Therefore the distance PF1 differs from the distnce PF2 by the radius of the circle. The two branches of the hyperbola are generated depending on which distance (PF1 or PF2) is shorter.
Moving F2 inside the circle transforms the difference into a sum, which yields an ellipse.