To answer this question, it helps to look really closely at the text, and especially at the very beginning and ending of the story.
Let's take a look at the ending of the story first, to get an idea of what probably happened to him after the story's events occurred. At the end, the police had torn down the wall and discovered the remains of his murdered wife; at this, we have to conclude that the man was arrested and taken to jail. Now, look at the beginning paragraph--in it lie confirmations of that assumption. The narrator states in that opening paragraph that he is going to tell us a story, that he really wants us to believe. Why is he so motivated to tell the tale? He states,
"Yet, mad am I not—and very surely do I not dream. But tomorrow I die, and today I would unburthen my soul."
He states that he is going to die tomorrow, and hence wants to unburden his soul from the weight of his awful act. He wants the entire world to know the true story, the awful details of the event, even if they do sound strange. He wants the truth out before he leaves this world. So, why would he be dying? Most likely, he is in prison, and received a sentence of death, and he is condemned to die the next day. That is the logical conclusion that one can draw from the hints given. So, the narrator is telling his story from the walls of his prison, on the day before his execution.
I hope that those thoughts helped to clear things up! Good luck!