Why does ice reduce swelling?
Swelling is one of the signs of inflammation. Inflammation involves release of histamine by mast cells present in the tissues. Histamine causes vasodilation and leads to leakage of fluid from the blood, along with which neutrophils and other WBCs also enter the area. They phagocytose microbes that might have entered with the injury.

Applying ice would cause vasoconstriction* (i.e. narrowing of blood vessels.) which would reduce the leakage of tissue fluid and hence swelling. 

*If you wonder why ice(or any cool thing) causes vasoconstriction think why you turn pale in winter ? Vasoconstriction reduces blood flow to the particular area and hence the exchange of heat is reduced which conserves body heat in cold environments.