The climax occurs when the criminals are caught robbing the bank.
The climax of a story is when things change, or the most exciting point. In a mystery, most of the time is spent trying to find the criminal, so the climax would be the moment when the criminal is caught.
A shopkeeper named Jabez Wilson, who happens to be overweight and have red hair, comes to hire Holmes to find out what happened to his job with the “Red-headed League.” Holmes deduces that his assistant is suspicious, because of his behavior and Wilson’s description, and they go to his shop to determine what is going on.
Once Holmes is there, he notices some things about the assistant. The “knees of his trousers” give him away as planning something in the basement (where, kneeling, he would get dirt on his knees). Holmes tells Watson that he is the man.
“Smart fellow, that,” observed Holmes as we walked away. “He is, in my judgment, the fourth smartest man in London, and for daring I am not sure that he has not a claim to be third. I have known something of him before.”
Of course, he is John Clay, “the murderer, thief, smasher, and forger”. He came up with the creative plan of robbing the bank next to the shop via the basement. Holmes figured this out. The only thing to do was wait to catch them in the act. When Clay and Archie break out, Holmes, Watson, Mr. Merryweather of the bank, two inspectors, and an officer are waiting. Holmes has it all planned out. This is the exciting climax.
“It's no use, John Clay,” said Holmes blandly. “You have no chance at all.”
“So I see,” the other answered with the utmost coolness. “I fancy that my pal is all right, though I see you have got his coat-tails.”
“There are three men waiting for him at the door,” said Holmes.
It works quite well. Holmes stops the bank robbery, and catches the robbers. He solves the mystery of the “Red-headed League” too. In the end, though, poor Wilson is the one who lost his £4 a week job!
Watson is impressed, once again, that Holmes could deduce so much of the case with so little information. He tells Watson that he is just happy that the case stopped him from being bored. This case is a perfect example of how Holmes's mind and memory help him solve cases quickly, and see a case where no one else would. He knew the story of the "Red-Headed League" was just odd enough to follow up on.