Stanley has been getting increasingly angry and annoyed at Blanche's presence in their small apartment and the airs she puts on of being superior to him. He does some digging and comes home to tell Stella of what he has discovered.
Among other things, Stanley learns that Blanche lied about taking a temporary leave from her school because of her nerves. In fact, she was fired because she had an affair with a seventeen-year-old student, whose father understandably became upset:
A seventeen-year-old boy—she'd gotten mixed up with! ... The boy's dad learned about it and got in touch with the high school superintendent.
According to Stanley, Blanche is more or less run out of town.
This is consistent with Blanche's poor decision making and tendency to live in a fantasy world. From what Stanley finds out, it seems that Blanche increasingly unraveled in her last years in Laurel after she lost Belle Reve. She became promiscuous, clinging desperately to whoever she could find to take care of her for a short time before finally showing up at her sister's home in New Orleans as a last resort.
While Blanche, in the tub, sings a song saying that dreams could come true if only people believed in them, Stanley, the representative of the hard, fact-based brutality of the modern world, reveals he has demolished the last of Blanche's hopes by telling Mitch about her past.