Horses trained into the higher levels of dressage are often ridden in a double bridle because the two different bits allow for more complex and refined rein aids, and the bridoon (snaffle) and curb have different actions - while the snaffle is good for direct aids and direction etc, the curb can be used to develop more elevation and precision in the movements.


Most dressage horses, even at the very top (Valegro for example) are ridden and trained regularly in a snaffle at home - a horse should not be ridden in a double bridle until it can perform all the same exercises in a snaffle. The double bridle just adds the finesse and enhanced communication that can give a performance the winning edge.

 

Up to second level you have to ride in one of the dressage legal snaffle bits and at 3rd and 4th level you have the option to choose to ride in a snaffle or a double its only when you reach the higher levels that the double bridle becomes mandatory
I think the reasoning lies in the origins of dressage which roughly translates as 'Training' and the tests were to test that training. Traditionally a horse was expected to initially perform correctly in a snaffle bit and then as it progressed it was expected to perform correctly in a double bridle
The UK has the same tradition in its showing classes where novices and lead rein & first ridden ponies are shown in snaffles and when you go into the experienced level classes you're expected to show your pony or horse in a double bridle or a Pelham with two reins

 

For FEI classes you're required to use a double bridle which is PSG, Intermediare 1, Intermediare 2, and GP. However you can show above PSG and all the way to GP in a snaffle for USDF/USEF classes. One of my trainers showed GP in a snaffle.

Also most well trained FEI horses school in either a snaffle or a double. They should be able to school in a snaffle. The curb does not create collection. The curb however does help with elevation and softness. If you have a horse that is very strong in the bridle and some are, even if they've been well trained. It helps lighten them off the bit and help them back off the bit. Some of the strong, heavy warmbloods are like that. It also can help with elevation but should not be the source of it.

The reason a double is used rather than a snaffle is partially tradition and partially for refinement. It is a skill to use the double bridle correctly. The double bridle is a tool, nothing more, nothing less. It can be used well and improve the horse's way of going or used appropriately or it can be abused. Just like any other tool. It's dependent on who is using it.

I know for my barn we introduce the double bridle at 3rd level. It doesn't mean we always ride in one but we introduce it to the horse and rider to use it. We usually introduce it for the first few rides with the curb rein loose and use the snaffle while the horse gets used to the two bits and gradually incorporate it, so they can get used to it.

 

For sure, internationally a full bridle is required (psg/i/gp), but not nationally. As far as why: each bit has a specific purpose and effect upon the entire body (snaffle=up/open/ifV/mobilizing the jaw, and curb closing/lowering/nuance).