The purpose or mission or goal of the "Red-Headed League," or "The League of the Red-headed Men," is not explained directly by Sir Arthur Conan Doyle, nor is it explained directly by John Clay posing as Vincent Spaulding, nor by Sherlock Holmes, nor by Dr. Watson. It is explained by Jabez Wilson to Sherlock Holmes as Wilson got it from Vincent Spaulding; and what Wilson told Holmes he heard from Spaulding is being written in the story by Watson. So much may have been lost in transmission. According to what Watson and Holmes understand from what Wilson tells them was explained by Spaulding:
"As far as I can make out, the League was founded by an American millionaire, Ezekiah Hopkins, who was very peculiar in his ways. He was himself red-headed, and he had a great sympathy for all red-headed men; so when he died it was found that he had left his enormous fortune in the hands of trustees, with instructions to apply the interest to providing for easy berths to men whose hair is of that colour. From all I hear it is splendid pay and very little to do."
If an American millionaire leaves a fortune to found any kind of trust in London it should be fairly big news. But Doyle explains why Wilson might not have heard anything about it. He has Wilson say:
"You see, Mr. Holmes, I am a very stay-at-home man, and as my business came to me instead of my having to go to it, I was often weeks on end without putting my foot over the door-mat. In that way I didn't know much of what was going on outside, and I was always glad of a bit of news."
Doyle has taken considerable pains to make Jabez Wilson into the type of character who could be fooled by such a preposterous scheme as the Red-Headed League. He is naive, "not over bright," a "stay-at-home," but a man who loves money and who is so thrifty that he welcomed the opportunity to employ Vincent Spaulding at half-wages. What especially appealed to Spaulding and his henchmen (who we learn in a later story included the nefarious Dr. Moriarty) was that Wilson's shop was in the perfect location for tunneling into the nearby bank.
It might be suggested that Wilson's blazing red hair always made him self-conscious, and this led to his becoming shy and reclusive. Another interesting thing in the story is that when Holmes first meets Wilson at Baker Street he says:
"Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else."
Holmes explains that he knew Wilson had spent time in China because of a Chinese tattoo on his wrist and a Chinese coin on his watch chain. Doyle often uses these examples of Holmes' deductive prowess to show off the great detective's intellect. But the fact that Wilson had evidently been living in China would provide another reason why he might not have heard anything about the formation of a League of Red-headed Men in England.
Wilson is completely taken in by the story of this Red-Headed League because he is shy, reclusive, unintelligent, enchanted by the opportunity to earn four pounds a week for doing very little (a typical clerk earned one pound a week in Victorian times), and because he may have been living in China when the death of the American millionaire and formation of the League had supposedly made the English newspapers.