Ammo manufacturers have told me that powder can be damaged by prolonged exposure to "high" temperatures, such as might be reached in garages or sheds in the south or southwestern states (100 degrees, plus...), and primers and powder both by exposure to solvent fumes. Apart from corrosion issues, there is apparently no concern about humidity, or low temperature storage (although powder ignited at low temps will produce lower pressures than that started at higher temps --- probably only significant to long range rifle shooters...) From what I've been told by the people who make modern ammo, "keep your powder dry" is no longer the operative admonition --- "keep your powder cool, and fume free" is the contemporary caution.
One good reason to keep ammo in its original packaging is to write its acquisition date on the packaging, so as to be able to use a "first in, first out" rotation, i.e., using older ammo first, especially important if you're stockpiling large quantities of similarly packaged ammo.