The long introduction to Dupin's method serves to set the stage and establish Dupin as a truly remarkable individual. It serves to illustrate Dupin's remarkable mental acuity and his ability to solve seemingly unsolvable puzzles. By introducing Dupin through his skills in chess, whist, and analysis of Chantilly's performance, it also serves to illustrate to the reader how Dupin will use those same skills to solve the murder in the Rue Morgue.
If the story had jumped right into the murder plot, the reader would not have had the opportunity to appreciate the full extent of Dupin's abilities. It would also have been more difficult for the reader to understand how Dupin was able to arrive at the solution of the mystery.