Really, there is only one reason, and if you were an old enough fogey you would know it from the ads: it hides the shadow behind the subject in the typical "up against the wall and say 'cheese'" picture.
One needs to keep in mind that these things arose at a time when a big "potato masher" flash might have a guide number of 45 (ISO 100 in meters, or 150 in feet), your film was going to be either VPS (ISO 160, but almost everyone shot it at 125) or VPH (ISO 400, but effectively 320 speed), and you were typically shooting at f/8 or so (because medium format). If you were using a small speedlight, guide numbers in the 28-30 range were more typical. (If you were shooting 35mm colour, ISO 400 film wasn't really an option.) Oh, and you may have had to gel your flash to tungsten and use an 80A filter on your lens, losing a whole bunch of light in the process (about 3 stops of effective flash range). Bounce was a possibility, of course, but because of the low film speeds, lower-power flashes then versus now, lack of post-shooting white balance and restricted apertures, that possibility wasn't always a practicality. Direct flash was the rule of the day.
That generally meant having a very visible and sharp head shadow, usually with the top of the shadow at about the subject's ear level, and because the waist-level viewfinder meant that a direct mount of the flash put it to the side of the camera, a little to the side as well. The bracket didn't do a whole lot about the quality of the light (it could be argued that elevating the flash made a difference in modelling, but all it did, really, was to widen the chin-strap and "Hitler moustache" shadows and hollow the eye sockets a bit), but it did force the head shadow on the wall lower and centre it behind the subjects. If you were close enough to your subjects and your bracket was high enough, the shadow would be completely hidden.
Yes, that sounds crude, and it was. It was a simple way of making "pictures of record"; the "real" wedding pictures were generally done under happier circumstances in studio. In this day, when wedding photography is a substantial pursuit and legitimate artistic endeavour, it may be easy to miss the fact that not very long ago it was a just dirty job that someone had to do. Between lugging around the better part of 100 pounds of gear while looking the very picture of calmness and coolitude while wearing a respectable suit (or a tux or morning coat), dealing with the limitations of the equipment of the era and managing what is still a rather chaotic scene, most photographers considered things like weddings a bump on the road to becoming a "real" pro. There were a few weirdos who seemed to enjoy the genre (Monte Zucker comes to mind), but then they were exceptions.
The brackets still have their uses today, especially if you're working solo and you want to use a large-ish softbox while roaming at an outdoor evening event as a main light. But there are much better ways of handling the original use case now.