Here's how the climax unfolds:
1. Holmes and Watson investigate the case of the Red-Headed League, which is a mysterious organization that recruits only red-headed men for a seemingly pointless job of copying an encyclopedia.
2. Holmes starts to suspect that the League is a cover for something more sinister, and he gathers clues that point to a potential bank robbery.
3. Holmes discovers that the man who runs the League, Mr. Jabez Wilson, is a pawn in a larger criminal scheme. Wilson's red hair was the key factor that led to his involvement in the League.
4. Holmes arranges a visit to the bank with Wilson and Watson to see if his theory is correct. During their visit, they observe suspicious behavior and realize that the bank is indeed the target of a robbery.
5. As the robbery is about to take place, Holmes and Watson confront the criminals, leading to a dramatic confrontation and the arrest of the criminals involved in the plot.
6. With the Red-Headed League exposed as a cover for a bank robbery, the climax of the story brings together all the threads of the mystery and reveals the true intentions behind the seemingly innocent organization.
In summary, the climax of "The Red-Headed League" occurs when Holmes unveils the criminal nature of the Red-Headed League and thwarts their plan to rob a bank, resulting in the arrest of the criminals.