The writer means that his grandmother was born as the Kiowas were leaving behind their glory days as a relevant, powerful Indian tribe in the Southern plains.
So, by the time the writer's grandmother was born, the supremacy of the Kiowas was in question. With the Comanches, the tribe once dominated the open ranges from "the Smoky Hill River to the Red, from the headwaters of the Canadian to the fork of the Arkansas and Cimarron." The writer maintains that the Kiowas were warriors and one of the "finest horsemen" the world had ever known. Yet, the Kiowas had one weakness: they did not view war with the same single-minded preoccupation as the United States Calvary.
The writer asserts that the Kiowas were ambushed by the formidable advance of the United States Calvary. Hedged in on every side, the Kiowa warriors had no choice but to surrender to the soldiers at Fort Sill. In the end, the Kiowa warriors were imprisoned in an old stone corral, which eventually became a military museum. The writer's grandmother was spared "the humiliation of those high gray walls by eight or ten years."
When the writer's grandmother was seven, she participated in the last Kiowa Sun Dance. The dance was held in "1887 on the Washita River above Rainy Mountain Creek." When she was ten, she witnessed soldiers from Fort Sill dispersing the tribe even before a Sun Dance could begin. The date was July 20th, 1890. So, when the writer's grandmother was born, the Kiowas were living the remnants of their proud, ancestral heritage. In the end, it was only ten years after the writer's grandmother was born that the Kiowas ceased to be a tribe altogether.