Why was the league created in "The Red-Headed League"?
The Red-Headed League was created by master criminal John Clay to trick Mr. Wilson into leaving his shop for several hours during the day so that Clay and his partner could tunnel into a nearby bank.
Mr. Wilson hires Sherlock Holmes to investigate the Red-Headed League because the office he was working out of suddenly closed. He had not worked for the League long. His job was a kind of sponsorship, because he had fiery red hair—or so he was told.
It turned out that Mr. Wilson had a very intelligent assistant, recently hired, who worked for less than the job should pay. He was the one who told Mr. Wilson about the job. As soon as Holmes knew the story, he figured it out.
“…[It] was perfectly obvious from the first that the only possible object of this rather fantastic business of the advertisement of the League, and the copying of the Encyclopaedia, must be to get this not over-bright pawnbroker out of the way for a number of hours every day….”
The description of the assistant was particularly interesting to Holmes, because he recognized him as a brilliant criminal. Holmes took the case to test his theory, and soon caught the would-be bank robbers red-handed as they tried to use the tunnel.