“To talk a blue streak” means to speak rapidly and without stopping.
The Oxford English Dictionary says that the phrase originated in the US in the early 1700s and that it implies a flash of lightning in speed and vividness.
It may “imply” vividness, but I’ve heard a lot of fast talkers whose words simply blur into mind-numbing babble.
The word “blue” is often added to phrases, sometimes for emphasis or rhythm, but more often there’s a story behind it. For example, “blue chip” as in blue-chip stocks, comes from the highest denomination of chips in poker, and means “the best.” Another phrase, “blue blood,” also means the best, or highest, social class, but it arose as a reference to white skin, which is transparent enough to reveal the blood-carrying veins beneath it.
“Blue in the face,” according to The American Heritage Dictionary of Idioms, means being exhausted from anger strain, or other great effort. It alludes to the bluish skin color resulting from a lack of oxygen, which might result from talking until one was breathless.
Other examples are: “the blues” to mean sadness, or a style of music, but there seems to be no connection between the two subjects. “Blue-collar” workers came from the blue denim shirts that factory workers used to wear.
We don’t know whether “swearing a blue streak” has different origins from “talking a blue streak.” Some believe that the word “blue” here refers to bad language. However, swearing became part of the expression by 1847, as in a “blue streak” of oaths.
The titles Blue Streak or Bluestreak have been used for: music albums, comics, films, roller coaster rides, fictional robot superhero characters, video games, a British ballistic missile, call signs, a bus network, a rubber company, a New Zealand railcar, and a species of wrasse, which is a fish. Wikipedia says there are 600 species of wrasse.
A similar phrase, “blue blazes,” was originally a reference to the fires of Hell, where it is said that brimstone burns with a pale blue flame.
So, what in blue blazes am I going to do with all those wrasse?