Double bridles are used optionally at Third and Fourth Level, and are a requirement for the levels above.
The purpose of them is to give the rider lighter, refined aids when they use their hands, and it gives more control over the placing of the horse's head (this does not mean holding his head in place!).
A rider who is using a double bridle should have an independent seat, leg, and hand. They should know how give and take at the right time (feel), and have very, very soft and elastic hands.
Overuse of the curb causes the horse to draw back his tongue (sometimes getting it stuck in between the two bits), and this is why some riders, especially the rollkur users, will put crank or other "trapping" nosebands on their bridles. It appalls me to no end to see the highest level riders pulling and yanking their horses around in the warm-up to pull them into hyperflexion. They end up, really, drowning the horse in pain and confusion.
That being said, their are many riders who do use the double bridle correctly. In the hands of a soft and empathetic rider, it's a great bridle!
On a side note, I ride Freddy in a pelham. It's an honest disaster to try a ride him in a snaffle, but I can't ride him in a double bridle (he will never be soft enough to go in one). I wish they hadn't made the pelham illegal in dressage competition, because it really get riders used to the double reins and teaches them some of the mechanics of a double bridle. The funny thing is that the pelham was originally created for dressage as a transition bit for horse and rider!