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Two kinds of argument can be made. One is thermodynamic - the gibbs free energy of the gamma/fcc/austenite phase drops below that of the alpha/bcc/ferrite phase at some temperature. That's pretty abstract but is what a metallurgist would tell you.
The other argument is a more intuitive one but needs some quick calculations. In short, the iron atomic radius can increase more in the FCC structure without reducing the density too much. This is because the FCC structure is actually a close packed structure in disguise - think stacking cannon balls - and these structures have the best possible packing fraction (~74%). If you calculate the iron atomic radius in the alpha/BCC phase it is 0.124 nm, compared with 0.126 nm in the gamma/FCC.
RefEdit: that being said,
@snottyotter might be right too. The BCC alpha ferrite is ferromagnetic and that will lower its gibbs free energy. I guess ideally a true metal would be FCC/CCP (cubic close packed) to increase density but a guy with more expertise than me points out here that transition metals aren't actually true/perfect metals and (presumably) have some covalent character to their bonds. That may prefer a slightly longer bond length which negates the density increase available from an FCC structure, especially when you include the energy benefit of allowing ferromagnetism.
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hamrack
@HHC
Can anyone explain how/why (why probably being the most useful) gamma phase iron and alpha phase iron change in their structures (FCC to BCC) due to temperature changes.