There are a number of studies which show that applying ice (cryotherapy) helps reduce swelling. This link deals specifically with applying ICE (Ice, Compression, Elevation) to ankle sprains. Evidence-based first aid courses such as those taught by the Red Cross all agree that using ice should be the first action taken when dealing with certain injuries.
From the previous document:
The therapeutic use of cold produces many physiological effects. Cryotherapy is
commonly accepted as the standard treatment in the management of acute injuries in order to
However, inflammation is a complex set of events that only becomes
more complex when looking at the specific effects of cold on inflammation.
be described by its vascular events, chemical events, and cellular events since acute cryotherapy
affects all three of these.
Vasoconstriction due to cold therapy counteracts the vascular
events of inflammation, specifically vasodilation. Cryotherapy also affects the release of over
100 chemicals that mediate the inflammatory process through the chemical events of
inflammation. Despite the effects of cold not being researched for most of these inflammatory
chemicals, the release or activity of several of the key inflammatory chemicals is known to be
inhibited as a result of cryotherapy.
From my own experience as a paramedic and training as a paramedic and first-aid teacher: Vasodilation (the expansion of blood vessels) is a common effect in injuries such as sprains and can even occur to such an extent that the patient briefly faints due to a blood pressure drop. Ice causes vasoconstriction (the tightening of blood vessels) which reduces the effect vasodilation has. Vasoconstriction reduces the amount of blood lost internally by limiting blood flow and inhibits the forming of edema, reducing overall pressure on the affected area. Ice also has the obvious advantage of numbing the affected area, reducing the pain initially felt.
Note that ICE (Ice, Compression, Elevation) is typical for first-aid scenario's. If a patient has ICE applied and it does not help after the first 20 minutes of cooling down a physician should be consulted. This is also the reason for the 'controversy' surrounding ICE: certain types of injury and mechanisms of injury do not respond well to ICE and should be treated differently. A fracture or ligament tear for example will respond to ICE by reducing pain and swelling but - for obvious reasons - cannot be treated by ICE alone. ICE is not a magic bullet, it's a guideline to provide first aid and the outcome of ICE will determine if a patient might need to see a physician.