The oil is what holds the wheat, flour, salt, etc together during the forming (similar to a piecrust or biscuit recipe). I have a recipe for homemade triscuts which are both baked and very greasy. I’m guessing the “unique process” they speak of is they partially bake it, stamp the phony woven effect on it and cut it into squares, then bake it some more.
It’s the same basic process as shredded wheat, invented by a guy named Henry Perky, who had digestive problems, and thought shredded wheat easier to digest. In 1895, shortly after having started to manufacture shredded wheat, he got the bright idea of baking the stuff into crackers:
I should add that Perky originally wanted to market his cereal making machine, and made the shredded wheat only as a means of demonstrating it. He eventually figured out that people were far more interested in buying the cereal than the machine.