My understanding is that it is of American origin, and has to do with the mundane and uninteresting lifestyle which may be perceived of the white middle-class.
2. bland; conventional.
Random House Unabridged Dictionary
in a way that is regarded as characteristic of the white middle
Oxford Living Dictionaries
Check out the example sentences to get a feel of how it can be used.
Here is an explanation of the meaning at The Phrase Finder.
Pertaining to the US white middle classes.
The Phrase Finder
Billy Joel's famous 1983 song Uptown Girl features the phrase, where he talks about an uptown girl living in her white-bread world, here are the lyrics:
Link to lyrics in video
I'm gonna try for an uptown girl
She's been living in her white bread
As long as anyone with hot blood can
And now she's looking for a
That's what I am
It's interesting if you watch the video, a group of grease monkeys singing about a girl in her white-bread world who are working in what definitely looks like a rough part of the neighbourhood. You can obviously get the meaning intended by white bread here, though let's not forget this was 1983.
Here are two excerpts from an article from Huffington Post titled "How White Bread Became White Trash"
In truth, though, the pejorative “white bread” had earlier
antecedents. In the diverse ranks of 1960s counterculture activism,
the phrase had already come to signify everything bland, homogenous,
suburban, chemical, and corporate—everything that the counterculture
hoped to upend.
But, by the early 1980s, another usage had emerged. In this case,
“white bread” signified almost the opposite: not dull affluent
suburbia, but white trash. “White bread,” like broken-down trailers,
came to denote poverty of a white and rural kind—the world described
by residents of TV’s South Park as “a quiet, little, white-bread,
Podunk, white trash, redneck corner of the U.S.A.”
Huffington Post article
Also, you can look through this short discussion among users at The Phrase Finder giving their understanding of the meaning of white bread.
Thus "white bread" describes the bourgeoisie not only because its
members are racially white but also because they subsist on highly
processed commercial foods.
Phrase Finder discussion
NB: This is just the opinion of some internet user.
It's hard to tell whether today its primary meaning is related to white middle-class lifestyle, dull lifestyle, or as the Huffington Post article put it, to denote "white trash". There's no doubt that it can be used in a primarily racial way, as seen in your original quote. This can be said in light-hearted playfulness or as a hateful slur, depending on context.