Jack London uses flashbacks in "A Piece of Steak" to help the reader see exactly where Tom King had once been in his life. The flashbacks simply help the reader compare and contrast Tom King's former status with his current one.
As Tom King is making his way to his next fight, on foot, he remembers what it was like when he was considered the best in the world.
Big money--sharp, glorious fights--periods of rest and loafing in between--a following of eager flatterers, the slaps on the back, the shakes of the hand, the toffs glad to buy him a drink for the privilege of five minutes' talk--and the glory of it, the yelling houses, the whirlwind finish, the referee's "King wins!" and his name in the sporting columns next day.
This lets the audience know that Tom King was, at one point, used to glory and fame and all of the privileges that came along with it. The flashback shows Tom King once had people fawning over him, buying him food, and practically throwing money at him. When compared to what Tom King is going through currently in the story, it's a sharp contrast.
The secretary of the Gayety Club had advanced him three pounds--the loser's end of the purse--and beyond that had refused to go. Now and again he had managed to borrow a few shillings from old pals, who would have lent more only that it was a drought year and they were hard put themselves.
This quote shows Tom King's current status in life: penniless and borrowing money from any source he could. Tom King fell from grace, and he had to learn how to reconcile that fact within himself.