Of course, it is very difficult to know what was in the hearts of the various white Americans at the time that Chief Joseph surrendered in 1877. Different people would have been impacted in different ways. There were surely those who hated Indians (this was, after all, a year after Custer’s Last Stand) and were not moved by his speech. Those who were moved were presumably moved because it is a very poignant speech that can serve as something of a funeral oration for the entire Native American way of life.
By 1877, Americans would have felt that they had just about defeated the Indians entirely. The Indians were no longer a real threat. It was already possible to romanticize the “Wild West” and a way of life that was either gone or quickly going. For example, by 1877, Buffalo Bill had already been touring for years with an early version of what would become his famous Wild West show. What this means is that people would have seen Indians by this time as something of a relic of past times.
Because of this, they would have felt pity and some degree of sorrow at Chief Joseph’s words. They would have recognized that these were the words of a man who wanted the best for his people but who was having to give up hope of living in freedom. With Indians no longer much of a threat, people could start to see them as real people and could start to feel sorry for them. This is why this speech was moving to many Americans.