The Nephites had been taught since the time of Nephi that their God would rise with “healing in his wings” (2 Nephi 25:13). That phrase, and idea, is best known for Malachi 4:2, but Jesus will repeat it later in 3 Nephi 25:2. Perhaps for that reason, the people understood that healing was associated with the Messiah, and that would be what Jesus perceived.
He invites them to bring those who are sick to him. There are very few populations of any size who do not have some who are ill, or lame, or blind, or are otherwise afflicted among them. In the ancient world, before modern medicine, it was even more common. Therefore, there were, in the congregation, those who could be healed.
They came, and they were healed. Understandably, their thankfulness, their adoration, was so great that “they did bathe his feet with their tears.”