This disparaging term refers to the supposed bland and uninteresting nature of white middle class culture in the US. It came into vogue in the 1980s and began to appear in print then. For example, this piece from The New York Times, 1981:
"Vincent... is white bread in a three-piece suit."
The term is virtually limited to the USA. In the 1990s the UK comedian Harry Enfield did his own take on 'white bread' when he created the obnoxious stage Yorkshireman George Whitebread - "I'm From Yorkshire. I say what I like and I like what I bloody well say". The comic device was modelled on Barry Humpries' dissolute creation Sir Less Patterson, the Australian 'Cultural Attaché'. Whitebread was a spectacularly unsuitable 'creative director' for an advertising firm.