Considering that it was Poe's objective to "entertain" his reader, but most importantly, to work toward a preconceived "effect," and then use words and symbols and language to help the reader reach or obtain that effect, this is a fascinating and appropriate question. Here is my response to each of your questions:
How does this story make me feel?
-Well, Poe was the inventor and master of detective fiction. As such, his objective was to make the reader feel a sense of suspense, initially, as he set up the problem that needed to be solved, and then to feel a sense of elation as his detective, in this case, Sir Auguste Dupin, solves the puzzle. When we discover how inept the Paris police are, I think we feel pity for them. When we learn how superior Dupin is to those around him, we almost feel a sense of superiority, as he does. When the story ends and we discover, as does the prefect and Dupin, that the letter was right in front of them the entire time, I suppose in some ways we feel a little stupid that we couldn't figure out the puzzle either.
What does this story make me think, and why?
-This story makes me think that any crime can be solved, provided those who are charged with solving the crime, don't overthink the crime. Most criminals are stupid, and the reason we know this is because most criminals get caught. Beyond this, however, is the reality that in life, most solutions are simple, and that nothing is as complicated as initially seems. That is the great lesson of this story, and others by Poe. A solution that seems impossible, ultimately becomes not only probable, but likely, once solved by the detective, or, in this case, the brilliant mind.
Hopefully this response helps you as you consider your own answer to these questions. Good luck!