This is not really a legal question. There isn't a "recusal register" on file anywhere where a judge needs to set down an explanation of his or her recusal.
The best way to find out is usually press reports from the time of the decision. Reasons for recusal are not usually secret, and press reports usually explain recusals in cases that matter. If they don't, though, the justices aren't required to explain themselves.
For the record: Justice Brewer recused himself from Plessy because of the death of his daughter. And it's generally accepted, although I don't believe any of them ever stated it publicly, that the three justices who recused themselves from Shelley v. Kramer did so because they owned property governed by restricted covenants prohibiting them from selling it to African-Americans (the subject of that case).