One simple model which seems to fit the "WWII ending the depression" piece of data (and which might have some overlap with the truth) is that it's relatively difficult to put idle resources into use, and significantly easier to repurpose resources that have been in use for other uses.

During the depression, a bunch of people were unemployed, factories were not running, storefronts were empty, etc. According to this model, under those economic conditions there were significant barriers to taking those idle resources and putting them to productive use.

Then WWII came and forced the country to mobilize and put those resources to use (even if that use was just to make stuff which would be shipped off to Europe and the Pacific to be destroyed). Once the war was over, those resources which had been devoted to war could be repurposed (with relatively little friction) to uses with a much more positive effect on people's standard of living. So things became good according to meaningful metrics like living standards, not merely according to metrics like unemployment rate or total output which ignore the fact that building a tank to send to war isn't valuable in the same way as building a car for local consumers.

The glaring open question here is why there might be this asymmetry between putting idle resources to use and repurposing in-use resources. Which is closely related to the question of why recessions/depressions exist at all (as more than momentary blips): once a recession hits and bunch of people become unemployed (and other resources go idle), why doesn't the market immediately jump in to snap up those idle resources? This article gets into some of the attempts to answer those questions.

(Bonus answer: World War One did not happen during a depression, so mobilizing for war mostly involved repurposing resources which had served other uses in peacetime rather than bringing idle resources into use.)