Marriage is a formal bond between two people, who promised to love and have a relationship with each other to the exclusion of others. An"affair" in this context is a relationship between two people by definition outside of wedlock, and is often the term applied when someone is breaking their marriage vows and having a relationship with a third party i.e having an "affair". Therefore the author is being ironic by calling "marriage a private affair" as it is not an " affair " in the above context at all, and using the other sense of the word affair justified by the word private. 
Hope that helps :)
