This is my first post on the forums, and I have a distinct lack of megaprojects under my belt, so I might be missing something obvious. I believe I understand the piston part of this, but I am not sure how bridges hold water above the displaced magma which is then used to cast obsidian. Can someone explain this?
Have you looked at the Bravebridged map posted in the OP? Another link to it
here if you need it. That shows a repeating piston design, though it's a bit hard to tell what it does if you don't know what you're looking at.
Anyway, on that map on level 174, there's a structure with 2 bridges, with a pump off to the east (this point of interest is marked "Old obsidian caster"). It's stoppered up by ice at the moment, but when it thawed, if the pump was run, water would flow out onto the bridges and be held there. A lever is linked to the bridges, and when pulled, those bridges retract and disappear, allowing the water to fall straight downwards. These bridges can be any distance above the top of the piston, though having a clear z-level between them and the piston is probably a good idea so you can do other work in there.
If you go a few levels down from that (PoI marked "Magma removal thing). At the north end of that you'll see a screw pump, separated from the magma piston device by some walls. That pump is part of a pump stack that, ultimately draws from level 167, which is the lowest level magma will be on after the piston is dropped. After the piston is dropped, the pump stack is run, drawing a specific amount of magma out of the piston space, so that there is exactly 2/7 magma on every square of the piston's top. The pump stack is then deactivated, the lever activating the water bridges is pulled, and water falls down. When the water hits the 2/7 magma, both the water and magma disappear to form obsidian. The process is then repeated twice more.
The most important part is probably that the repeating design requires that you put the support at the bottom of the piston, not at the top as the pictures show. If you go to the "Magma cistern" point of interest on the map at level 17, you can see a single tile of magma that is darker red. This indicates that there is a building there, specifically the support. It must sit on some floors that are invisible in the magma. Those floors would get annihilated when the piston fell on them.
The bridges really just need to be above the magma, with open space beneath. When a bridge is retracted or raised, it doesn't fling water anywhere, so any water on the parts that just simply disappear just simply falls straight down. So, there's not really much to explain.