Exposition provides us with important background information about the characters that will become significant later on. In "The Ransom of Red Chief" we're introduced to the hapless kidnappers, Bill and Sam, as well as their intended victim, Red Chief, or Johnny. The very first line tells us that all did not go according to plan concerning the kidnap plot:

"It looked like a good thing: but wait till I tell you . . . "

Rising action relates to what happens in a story after the exposition and before the climax. Here the main problem is established and how the relevant characters attempt to solve it. In "The Ransom of Red Chief" the problem is that little Johnny's driving his captors mad with his brattish behavior. It's not a problem that Bill and Sam could reasonably have predicted and so they don't know how to deal with it. At this point in the story, we're starting to wonder how this problem might eventually be resolved.

The solution to the problem comes in the climax. In also provides the turning point of the story, that moment when the characters experience a significant change. In this particular case, the incompetent captors receive a letter from Johnny's father telling them that they need to pay up to return the boy. They are mightily relieved to think that they may have finally found a solution to their problem.

The falling action occurs after the climax. Bill and Sam return Johnny to his father. They're glad to see the back of him, but Johnny's rather sad; he was so much happier with Bill and Sam than with his father. To him, his kidnapping was all just a big adventure.

The resolution is that part of the story where everything comes together and all the various problems and conflicts are finally resolved. A wailing Johnny doesn't want to let go of Bill's leg, but his father just about manages to pry him off, allowing Bill and Sam to make good their escape.

The exposition is the background on Johnny Dorset and the kidnappers, the rising action includes, in my view, the kidnapping of the boy and the two men locating to the remote cave. 

The actual climax of the story occurs when the kidnappers are forced into submission by Johnny and a reversal of power takes place. 

"During Sam's absence, the captor and the captive change roles, seemingly only in play but actually in real control of the situation. When Sam returns to the cave, he finds Bill, badly battered, playing the captured trapper to Johnny's heroic Indian, who calls himself “Red Chief.” Appropriating Sam for his game, Johnny announces that Bill is to be scalped and Sam burned at the stake." 

During the confinement, Johnny abuses his kidnappers thoroughly, having a good time with his new playmates.  The falling action occurs when the kidnappers make contact with Red Chief's father, looking for a ransom.

The resolution of the story is a reversal or a twist, very common in O Henry's work.  The kidnappers, Bill and Sam must pay Johnny's father a sum of money so that he will take the boy off their hands. 

"Johnny, however, does not wish to leave his captors. They must scheme to get him back to his father as once they had schemed to get him away from home, and finally, they must run at top speed to escape the boy who does not wish to lose his new playmates, the would-be kidnappers who have become his victims."