Why do batteries discharge faster in cold air?

Answer 1

They don't. It used to be recommended that batteries be stored in refrigerators to prolong their lives. Additionally, from chemistry, we know that chemical reaction rates increase as temperature increases, not the other way around.

What is possible is that whatever you're powering with the batteries might consume more power when it's cold. CMOS devices run faster at cold, thereby potentially consuming more power at cold. Or, there may be some internal heater circuit that draws power when it's cold, but doesn't when it's hot.

Answer 2

Vehicle batteries have two strikes against them even before you turn the key on if its a cold morning.
First, the engine is going to offer much more resistance to turn - and the battery has far less output energy due to the slowdown of the chemical reaction inside the electrolyte which produces the current.
These two factors will run the battery down much quicker than it normally would on a warm summer morning.