How do the three hunters advance the plot of the story?

Essentially, the three hunters advance the plot of the story by appearing at a most opportune moment: right after Mrs. Sappleton's cheerful monologue about her husband and brothers. The hunters' demeanor, manner of dress, and actions appear to substantiate the details in Vera's story about the family's tragic history. The author uses the three hunters to highlight Framton's paranoia and Vera's manipulation of the easily influenced hypochondriac. 
In Vera's eerie story, she includes important details such as the brown spaniel, the youngest hunter singing a particular ditty, the "treacherous piece of bog," and the fact that the men were hunting for snipes. These details, of course, are validated when the men show up in the exact way Vera intimates.
Additionally, Vera's story about her aunt's trauma manifesting as mental instability is reinforced in Framton's mind when the men appear. As a result, instead of the "complete rest" and "absence of mental excitement" that the doctor ordered, Framton is subjected to a gothic story full of morbid details.
Vera's story is what inspires Framton's hurried exit by the end of the story. The three hunters advance the plot of the story by appearing to substantiate all the details of Vera's morbid story.