Its purpose is to increase mpg, and decrease emmisions. Now, will you see such gains, depends highly on your driving environment and driving style. Around here, not so much unless you participate in the Drive of The Lemmings morning and night coming in or out of base from the north at peak times.
Now, as to how it works, and this is based on the wifes Jeep, so your vehicle might vary a bit, but should be similar. On heres, there are I think 5 or 6 conidtions that must be met before the system even works.
1.Fuel above a certain point
2. Battery and charging system check good, including a "history" of good ebahavior from the battery. When I swapped the battery on hers, it took more than a week to gather that data.
3. If the heat or A/C wants more, they system either wont engage or will turn the engine back on as needed.
4. Brakes must be held down past a certain point, and leeting up restarts the engine (important point, note that)
5. Wheels must be pointed straight ahead
6. Must be in Drive, not park or reverse
Now, it does take some getting use to and change in technique at stops. What I'm in the habit of doing is in really hot or cold temps, turning it off. At stop lights, when the engine kicks off, I let it, until I'm close to getting a green light, based on other lights (left turn goes yellow, I'll get a green soon, etc), then I ease up on the brake until the engine starts. I found that waiting and going right from brakes to gas result in choppiness. Maybe that's my lead foot, but it was annoying. When you enter a parking spot, either dont straighten your wheels or dont dawdle selecting park. Otherwise it will kick in and then when you do select park, the engine restarts, only for you to then turn the vehicle off. another annoyance.