The wall studs are obviously taking in a lot of water. Once they're all dry, poke into them with an ice pick to see if it's just surface rot/mold, or if they're soft and no longer sound enough to support the walls. If that's the case, and I suspect it is, plan on removing and replacing all rotten studs, the bottom shoe plate, and anything else that's no longer functional. If the tops are all sound, just cut off and remove the rotten bottoms, and sister in some helper (partial) studs alongside the new replacement "stub" studs. Not a big deal, really, but you'll need to remove and replace all of the exterior sheathing and stucco as well. Still not a big deal. Make sure to use blue board or green board on the interior walls--NOT standard white drywall.
The important thing is to determine and eliminate the source of water getting inside the wall. If your shower surround above the tub was faulty, it's a very easy fix to properly install a new surround. But if it's the small roof over the bump-out that's leaking, plan to do what nealtw said and remove/replace it with something that's capable of keeping the rainwater out. And don't forget to install proper flashing under the adjacent wall's stucco.
