Well, it depends upon which version of the show you're using. In the US version, the song is given to Florence in the script. Freddie, the american chess player whom she is assisting, and her sometime lover, has just caused an international incident by accusing the Soviet player of cheating by way of receiving coded messages in cups of yogurt. He walked out of match and left her to clean up the mess, which she did by arranging a meeting in which the Soviets. When she confronts Freddie about the deed and tells him she's arranged a meeting, he refuses to attend without an outright apology. An argument ensues (The American & Florence / You Want to Lose Your Only Friend) and finally, Freddie storms out. Alone in the room, Florence sings Someone Else's Story as a lament to how ridiculous her situation is - she's putting up with an abusive partner and setting up meetings to justify his histrionics. She doesn't understand why she's in this position, but after seven years it's difficult to call it quits. Later in Act One she does leave Freddie, so this is really the beginning of that arc for her character.
In the UK version (and lots of variations of the US version, these days) the song is given to Anatoly's estranged wife Svetlana in Act Two. Her situation is a similarly ridiculous one - she has been brought to Bangkok (or Budapest, or wherever, depending on version) under duress purely to put emotional strain on her husband. She knows about his relationship with Florence, and also knows that her marriage to Anatoly has been over for some time now - but the two did indeed once love each other; some could argue that Svetlana indeed is still in love with him but her feelings are not reciprocated. The degrading nature of her situation as a political pawn, coupled with seeing her once and/or current love with another woman is the catalyst for the song in Svetlana's case.
In the Sydney 1990 production, the song's lyrics were changed slightly to better fit Svetlana, and while some productions choose to use these lyrics, most use the original, more familiar lyrics written for Florence whether Florence or Svetlana has the song. The 2008 concert with Josh Groban, for example, given the song to Svetlana but leaves Florence's lyrics intact.
In either case, it's the story of a woman in an absurd situation, one she'd never place herself in willingly, yet for one reason or another, here she is: trapped in a moment she cannot escape from, with no choice but to see the story through to its inevitable end.
Hope this helps!