(Definitions from OxfordDictionaries.com)
In the case of this story, "Imperial Grand Shrines at Ise" refers to the shrines of the emperor's family
In the case of this story, "Imperial Grand Shrines at Ise" are described as "the most sacred sanctuaries of Shinto"
In the case of this story, "Though the origins of this custom may be shrouded in mythology," the writer means that it is not clear how the custom started.
In the case of this story, "Though the origins of this custom may be shrouded in mythology," the writer is referring to the stories related to the Shinto religion and the Shiken Sengu tradition
In the case of this story, "the faithful adherence to the principles of the Shikinen" refers to the loyal and constant support of the principles of the Shikinen.
In the case of this story, "the preservation of ancient Japanese architectural and ritual forms" means that old Japanese building construction and design have been kept alive.
In the case of this story, "the preservation of ancient Japanese architectural and ritual forms" means that the actions of old Japanese ceremonies have been kept alive
In the case of this story, "As the ancestral shrines of the emperors of Japan" refers to the shrines of the emperor's family
In the case of this story, "symbols of the deities’ spiritual presence" refers to symbols of the Sun Goddess and the God of Agriculture