A joke, mostly within the tv show Family Guy, that is set up as a flashback and has very little, if anything, to do with the main storyline of the episode.
The cutaway scene about the Asian driver was very funny even though it did have anything to do with the story.
To cut an elastic strap typically referred to men or women underwear and fling the offensive smelling and saturated material away in order to prevent saturation of the pants.
When Niles shit his pants, he was left with no other option other than run to the restroom and perform a cutaway.