The story is set in Groveland, Ohio. The action takes place at Mr. Ryder's fine home.
There are several "themes" in this story. The story focuses on appearance v. reality, social life, racial identity/racism, and also the concept of personal identity. Chesnutt was a Realist (more specifically considered a regionalist) writer, and he was interested in capturing and presenting "real life" -- that is, the sorts of experiences that occurred, rather than fantastical or sentimental ones. Chesnutt was a very light-skinned Black man, who could pass for White if he wanted. At the time, passing for White would have given Chesnutt an easier and more comfortable life. However, he chose not to pass, and declared himself a Black writer/spoke candidly on Black issues.
As for the conflict/plot: Chesnutt's acknowledgment/celebration of his racial history is mirrored in the action of "The Wife of His Youth." Chesnutt's main character, Mr. Ryder, is a light-skinned black man living during the Reconstruction era (post slavery). His social circle is made up of other well-off, light-skinned African Americans, and he seems obsessed with color and with Whiteness. He wants to marry a light-skinned widow and eventually lighten the race at large. The underlying concept is that he wants to "erase" Blackness. This is about more than skin color; when Ryder wants to erase his Blackness, he's really talking about his past. The pain and degradation of slavery haunts him and makes up his cultural history. He views white as better and less painful/shameful; that is why it is so important that, at the end of the story, his chooses to acknowledge the wife of his youth. Through this acknowledgement, he is acknowledging his Blackness not as shameful, but as an integral part of him. The loyal woman he presents is a positive image of Blackness.