Try a simple, at home experiment. Blow up a balloon inside your house. It would be at room temperature of say, 72F. Put it inside your refrigerator for about 30 minutes, then take it out. You will find that it has deflated.
Locker room temperature 70 degrees, give or take. Field temperature 45 degrees, give or take. The same principle applies. The air pressure inside the ball will go down as it assumes the field temperature.
Supposedly, when this was discovered at halftime, all the balls were inflated to the legal pressure. First half, Patriot outscored the Colts 17-7, using the deflated balls. Second half they outscored them 28-0. Any advantage the Patriots gained would seem to be minimal. I guess the underinflated footballs made the Patriots block and tackle better.