The salt solution is wicked up into the cardboard tree via capillary action. Water evaporates from the surface of the tree, forcing the salt to crystallize out.
Mrs. Stewart's Bluing is a colloid -- tiny particles suspended in water (think of glitter in a snow globe, but much, much smaller). The tiny particles make it easier for the salt crystals to form.
The ammonia helps speed up the evaporation process, which makes the crystals grow faster.
There's a more detail explanation available from Mrs. Stewart's Bluing at http://www.mrsstewart.com/pages/explanation.htm