The motivation for CB is that most "big" stuctures are physically bult from subcomponents with which can be modeled separately. To make a reduced dynamics model of each subcomponent, you need to retain all the degrees of freedom at the boundaries which wll join to the other subcomponents. But if you eliminate everything else from the model, you lose the dynamic behaviour happening "inside" each subcomponent.
The basic idea of CB is to represent the internal dynamics of each subcomponent by its vibration modes with its boundaries fixed. This works well when the physical connections between the subcomponents are simple compared with the dynamics of each subcomponent, for example a satellite attached to its launch rocket at a small number of mounting points.
Another benefit is that each subcomponent model can often be checked against test vibration measurements indepedendently, which makes it easier to find and fix problems than working with just one large model. In fact the component models can be constructed directly from measured data, instead of making a conventional FE model.
There are variations on the basic idea - for example it is possible to the vibration modes of subcoomponents with the boundaries free to move instead of fixed (or with some parts of the boindary fixed and the rest free), which may improve the accuracy for a given size of model. Devising the "best" way (and even defining what "best" means) is an ongoing research topic.
This looks a pretty good summary of the math (it's slightly NASTRAN-flavored, but most of it should make sense if you don't know NASTRAN):
http://femci.gsfc.nasa.gov/craig_bampton/index.html
THe guutar analysis example here is fairly typical of the practical issues with this type of dynamics modelling: http://www.sem.org/pdf/substructuring_tutorial_imac2010.pdf