Question: What's the concept? Is it just that all these random songs are being played in a sort of "concert" given by the Hearts Club Band or does each song have a part of the story too?
"Themes and structure
With Sgt. Pepper, the Beatles wanted to create a record that could, in effect, tour for them — an idea they had already explored with the promotional film-clips made over the previous years, intended to promote them in the United States when they were not touring there.
McCartney decided that they should create fictitious characters for each band member and record an album that would be a performance by that fictitious band. The idea of disguise or change of identity was one in which the Beatles, naturally enough, had an avid interest — they were four of the most recognizable and widely known individuals of the 20th century.
The Beatles' recognition was the motivation for their growing moustaches and beards and even longer hair, and was an inspiration for the disguise of their flamboyant Sgt. Pepper costumes. McCartney was well known for going out in public in disguise and all four had used aliases for travel bookings and hotel reservations.
Thus, the album starts with the title song, which introduces Sgt. Pepper's band itself; this song segues seamlessly into a sung introduction for bandleader "Billy Shears" (Starr), who performs "With a Little Help from My Friends". A reprise version of the title song was also recorded, and appears on side 2 of the original album (just prior to the climactic "A Day in the Life"), creating a "bookending" effect.
The cohesive structure and careful sequencing of and transitioning between songs on the album, as well as the use of the Sgt. Pepper framing device, have led the album to be widely acknowledged as an early and ground-breaking example of the concept album."