Most gasses follow pretty closely to an equation called the Ideal Gas Law, stated here : PV=NkT
Where P stands for pressure, V is volume, N is the number of molecules in the sample, k is Boltzmann's constant, and T is absolute temperature typically expressed in kelvins.
For the situation with a basketball left outside on a cold day, the number of molecules in the ball, N, is staying pretty much constant. So when the temperature drops the product of Pressure and Volume has to drop. The Pressure, P at the point the ball drops to 1atm and starts to deflate is going to stay in equilibrium with the atmospheric pressure, so the only thing left to change to deal with the temperature dropping is Volume.
It should be noted that the air in the ball isn't quite an ideal gas, and the ball itself will be exerting some forces as well so it won't behave precisely as this equation states but its a pretty good approximation.
Thanks for the question,
(published on 02/14/12)