What are the advantages of having a off camera flash bracket?

The reasoning isn't red eye - it's the shadow. More so, the shadow casted across the face: pockets under the eyes, shadow below nose, etc.

Having an off-camera flash bracket allows us to provide a soft, even light over the subject providing a nicely lit subject.

Onto the red-eye debate - there is no scientific proof as to what causes red eye. THere are theories, assumtions, studies, but that's really all the further it goes. I can photograph directly into someone's eyes and get redeye about 8% of the time one time and 80% of the time another time. There isn't a definitive answer yet at this point in time (argue all you want, yer an idiot if you try).

If the flash comes from straight on, you're a lot more likely to get red eye and unattractive shadows behind the subject. A bracket helps eliminate that problem.

It could be that they are bouncing the light at a specific angle which in turn will make it look like the light comes from a source on the side or from above.

Eliminates red eye effect and can make it easier to subdue shadows.