A Doll's House is a realist play because its plot and main problem deal with a situation that, either at the time of the setting, or in the past or future, could actually occur.
In a Doll's house Nora battled the limitations placed upon her by society while she tried to save her husband. The limitations placed on making loans, doing business as a woman, and accepting money from a man, placed Nora in a situation of near-crime. Yet, she was not a criminal, far from it, but she did face the reality of an unfair society, of women's rights being limited, and she also experienced the ungratefulness of being expected to do too much for too little.
Her problem, and the way she solved it - by breaking with expectations and finding herself again- are quite realistic possibilities in everyday life. This is what makes it a piece of "realism"- precisely realistic fiction.