The climax in any story is the turning point in the story. It's the moment when the conflict has reached its greatest point of intensity and the moment when the resolution is in sight. The reader is no longer in the dark about exactly how the story will be resolved, regardless of if that resolution is tragic or triumphant.
In Daphne Du Maurier's short story "The Birds," the reader still has hope that the protagonist, the farmer named Nat, and his family will survive well into the last page of the story. In fact, the reader still has hope for survival the moment Nat informs the reader he had just had a brainstorm to "fix barbed wire in front of the boards" across the windows. It is after this paragraph that the reader finally understands all hope is lost. Soon, Nat hears "the tearing sound of splintering wood," as the birds attack the door, which tells him it won't be long before they gain entrance through the door. If the birds gain entrance, Nat knows he and his family have no hope of survival, which is why he asks for his last cigarette, similarly to how convicts on death row ask for a last cigarette and a last meal just before the execution.
Hence, the moment he hears the wood of the door splintering is the moment the conflict between him and the birds reaches its climax. All of the action from the moment he asks for the last cigarette to the end of the story is the falling action and resolution of the story.