The Impossible Conversation is, like its name suggest, an impossible conversation, or at least one that will probably not lead to a happy ending in its current form. You (an unnamed person) is having a conversation is your (probably former) best-friend following an unexplained conflict. Your choices in where you bring the conversation forward should influence how your relationship with the person go.
Due to the nature of the conversation, the writing is heartbreaking and painful. The end of relationships are hard, especially when people have strong bonds with one another, and it is the more painful when both parties have hurt one another (though, in this case, the hurt seems more one-sided). The bareness of the interface (keeping to the base Harlowe UI) and interactivity (click to show the next words) does add to the struggle and dreariness of the situation.