The 30 minute version of the cold water safety video.
This video shows the effects on people who fall into cold water. With water below 68 degrees it can take a swimmer 1 minute to catch their breath. They will then have about 10 minutes while they can still move, and in 1 hour they will lose consciousness due to hypothermia.
If the swimmer is not wearing a life jacket they may not make it through the first minute due to difficulty breathing, and will likely die before the 10 minutes have passed because they won't have the strength to keep their head above water. A life jacket makes survival much more likely.
It is very difficult to put your life jacket on while in the water. If you are not wearing it when you enter the water, it will likely be impossible to put on after you are in the water.
The life jacket will make the initial shock of entering the water much easier to manage and make the task of rescuing yourself much easier an more likely to succeed.
This video was made by The National Water Safety Congress: http://www.watersafetycongress.org/ and the US Coast Guard