The term was coined by Costikyan in Paranoia, a game wherein players get the equivalent of extra lives in a video game, because the game is so lethal. This is achieved by PCs being generated in clone banks, and there being additional copies made (every good scientist makes back ups). In the GM's section, they discuss the nature of the clone family in detail. Why, the text asks rhetorically, are all members of a clone family the same?
The text then provides the following:
The Real Answer
The Transparently Bogus Rationale
(three paragraphs of setting info providing a hard, setting-based reason why it must be so that is, however, transparently bogus and after-the-fact)
I think it's important for people to understand that a great deal of gaming involves TBRs. That is to say, we decide what is best for the game first, and then justify it. And when I say we I mean not just game designers but GMs too. At least, that's how I GM. Fiction is, after all, infinitely malleable, no matter how defined the setting.
But very often you'll see people trying to reason something the wrong way around. That is to say, they reason arguments based on TBRs, instead of just deciding what is best for their game first. Don't get me wrong, there's nothing at all wrong with going "well if the setting has X=Z as a fact, what does that mean about Y?"
The example, IIRC, was people trying to reason how the WFRP world could have arrived at a position which is rather less sexist than Europe of the 17th century. Some seemed to think that if that couldn't be explained, the game should be made more sexist. But you start with what's fun, and then provide the Transparently Bogus Rationale. Why are there dwarfs with mohawks? The TBR is that they are too ashamed to wear beards but cannot totally remove their hair blah blah blah. The real answer? Cos it's awesome.
Warhammer is built entirely on TBRs. There's so many of them and they're so ridiculously transparent that the thing resembles a china shop. Which doesn't at all mean the setting can't feel real, or deep, or alive. Nor does it excuse lazy writing - good TBRs are wonderful things. But be assured we writers know they're TBRs, and you should too.