The extinction of griffinflies, which were huge, dragonfly-like insects with wingspans of up to 70 cm, is usually attributed to the fall of atmospheric oxygen levels from more than 35% to levels below the current 21%. Insect breathing becomes inefficient for large bodies, but this can be offset somewhat in the presence of high oxygen concentrations. It's therefore not unreasonable for higher oxygen levels to have allowed larger insects to exist, and conversely for these to go extinct once the oxygen levels fell.
However, falling oxygen concentration is not the only event that roughly coincided with the extinction of the griffinflies. Until the evolution of pterosaurs insects had enjoyed a monopoly on powered flight, and it might have been this lack of competition that made such large insects viable. It is unknown exactly when griffinflies went extinct and when pterosaurs first evolved flight, but they may have briefly overlapped. If so, the extinction of griffinflies could have been due to competition from the first pterosaurs.
Finally, the largest known mass extinction in Earth's history, the Permian-Triassic extinction event happened right in the middle of the time-range during which Nel et al. (2008) estimate that the griffinflies went extinct. I have never seen this mentioned as a potential cause for their extinction, but it seems like a pretty obvious thing to consider.
Gnuplot-script used to generate the figure.
Disclaimer: I am not an expert on this. This is based on a few hours of literature searching.