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My high school chemistry is kind of rusty, so I'd like to be educated on how this works. I thought that O would bind more strongly to Fe than N. If so, then how would they get the N in the ammonium nitrate to displace the O?

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Indeed from the physorg article illustration (cited up top) and their description, the N doesn't seem to form a molecular bond with F, but to diffuse into the iron crystal; with F binding to F and N being trapped but not bound. Fe16N2 is then a description of the crystal ratios, not a description of a molecule that can create a crystal.

My chemistry is prob as weak as yours though, so don't trust this too much.

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