This has a nuanced answer. The short of it is he writes to someone he knows where she's from and he discloses Blanche's sordid history. But, there's a bit more to it than that. He makes the decision to do this because he thinks Blanche has wronged her sister and his wife Stella. Thus, he chooses to write the letter and disclose all that he's learned at her birthday dinner. He does this not out of love, but out of anger at perceived slights that he sees between Blanche towards Stella, and by proxy himself.