The way you can work out the bindings working, which seems extremely complicated, seems to be the following:
The bindings function by modifying the truename of a being (without having to know the truename of the entity), by sort of looping it back around on itself. However, if that was just the case, then it would re-assert itself once the ritual was complete.
The process goes one step further by creating a binding object created to be either made out of the deepest parts of the shadowfell, or has been aligned as such (which ties in with their ability to align materials to various planes.). The truename of the being is then “looped” around and through the binding stone or object, which then passes through the shadowfell, and ends up getting…for lack of technical terms, “knotted up”. The key factor here, preventing the reassertion of the being’s truename is that the nature of the deep parts of the shadowfell the name is looped through is suffused with depression and lethargy, sort of preventing it by removing it’s “will to act” (at least locally…sort of like paralyzing a limb without paralyzing the whole creature). Furthermore, because the truename is looped around and through this binding object, it ends up assimilating that object as a virtual part of the “self” of the being. If, then, the object were specially prepared with a binding agreement, the outsider would act as if it had agreed to the binding.
Often, keystones of buildings were used as the binding object, and the rest of the building was then specially prepared to be bound to the keystone so that the being’s “self” would essentially be the entire building. This has a number of added beneficial side-effects.
Obviously, this method doesn’t work for entities that are already tied to the shadowfell, as they are partially immune to this procedure.
The City here did much work in several different areas, competence of which led to their figuring out shadowbinding.
The first key technology was manufacturing so-called “aligned materials”. These are materials that are treated so that they have a planar alignment, which can have a wide variety of effects, depending on the plane so aligned. The major breakthrough, which took a very large amount of time and energy was the creation of the planar forges as they’re called. These are those series of rooms that you passed through to get here. It is through their usage and some rituals that one can easily align a material.
The second key technology was in truename modifying. Originally developed by the college, it dealt with temporary ritual modification to a being’s truename to give one an edge in bargaining and bonding. In general, the techniques were considered much too much work for very little, or not long-lasting effect, so were reserved for very rare times. This changed very much once shadowbinding was discovered, and a way to make these temporary changes more permanent was discovered.
In one of the documents you read, which seems to piece together the original creation of shadowbinding after the fact, you find that its creation was sort of shrouded in mystery. The members of the bardic college were rather secretive as a group to begin with, but this individual was even more than most. It was known that he more than dabbled in summonings and bindings, and that he often consulted with extra-planar entities. It is believed by the writers that powerful shadowfell entities must have been involved in developing the key component to shadowbinding (that of the properties and nature of certain deep parts of the shadowfell).
One thing of note, you find references that there might be some lore here key to working with Drown/etc… to remove them, returning them to the original primordial.