Scout, the young narrator of Harper Lee's novel, To Kill a Mockingbird, was born with the name Jean Louise Finch. Her mother died when she was very young and Scout has no memories of her. I don't believe there is any indication in the novel why, how or when Scout earned her nickname. I believe it is probably indicative of her curious nature--a scout being someone who searches out things ahead of others. It is also a boyish name, fitting for her tomboy personality.
Scout's real name is Jean Louise Finch. There is no place in the novel where Harper Lee describes why her nickname is Scout, but as the reader comes to know her, it becomes obvious that it is a fitting nickname for her.
Scout is a curious tomboy, and since scout's in real life need to be both tough and curious in order to do their jobs, this nickname makes sense for her.
Nicknames are common in the book and many characters, especially related to the children, have nicknames.
Scout's brother, Jem, is really named Jeremy Atticus Finch.
Boo Radley's real name is Arthur. Boo seems fitting for him as well, since he is basically the town's boogeyman and tales of him scare small children.
Dill's real name is Charles Baker Harris, but he smells like pickles, so he is called Dill instead.
Interestingly, in To Set A Watchman, the book told from Scout's adult perspective about her adult life, she no longer goes by Scout, but still refers to Dill and Jem by their childhood nicknames.