It is a way to keep your food warm and cooking properly during the period in which the hot water that you add to freeze-dried food does its magic. (You still need to carry a means to heat/boil water ... I use a Snow Peak Canister Stove because of its light weight and convenience, and you can recycle the canisters if you punch holes in them first using, say, a Jetboil Crunchit Recylcling Tool .)

Particularly if you repackage your food in freezer bags, the heat dissipates quickly, while it is supposed to be "cooking." But if you put your freezer bag (with food and hot water) in a cozy while you weight the 10 or so minutes for the food to "cook," you will end up with a warmer (and maybe more cooked) meal.

You can also use a warm hat as a cozy, except then it will smell like food, and you'll have to store it in your bear canister, instead of wearing it when you sleep. You can also make your own cozy. Here's one approach: http://sectionhiker.com/diy-freezer-bag-cooking-cozy/.

During the day, I put things that are prone to melt in my cozy (e.g., trailmix with chocolate chips), and they stay cooler. At night, after I use my cozy, I put it in the bear canister.

"Freezer bag cooking" is one approach to cooking on the trail that can help reduce weight and post-meal washing time. It is discussed here, particularly here. When you are done with your meal, you put everything dirty in the freezer bag, seal it up, and throw it in the bottom of your canister. The down side is that you do accumulate some trash (lots of plastic baggies), and also, long-term I'd prefer not to eat food that has been steeped in hot plastic. But, for occasional backpacking, I don't sweat it.