Chrysanthemums are not native flowers to Japan but they have been cultivated here since the 8th Century. The flower came to be adopted by the ruling classes and is of course the crest and official seal of the Emperor – hence the term Chrysanthemum Throne.
Chrysanthemum Festivals or Kiku Matsuri are held in many shrines, temples, parks and gardens during the 9th lunar month from around mid October to mid November. One of the more famous is held at Kasama Inari Shrine in Ibaraki prefecture.
These flower shows are still very popular so I am not sure what the caption of this postcard is trying to say.