To do this, you need a way of characterizing different structures or patterns, and one of doing this is to write down a procedure which will reproduce the pattern, to a certain level of abstraction and accuracy. Two histories, two series of past data, leave you in the same causal state if they leave you with the same distribution of future data, i.e., if it makes no difference to the future whether you saw one data-series or the other. This being the case, the difference between the series is unimportant, and we lump them together. This procedure identifies causal states, and also identifies the structure of connections or succession in causal states, and so automatically creates an automaton in the lowest Chomsky class you can get away with. We're trying to fit our data to models of pre-specified classes , and to find the simplest, most accurate model we can.
Dynamics is a bit trickier.
Notice that we're just using our data, not some a priori model of the process, to try to say what the underlying mechanisms are like. It happens that for some dynamical systems one can prove that by the right coarse discretization you doesn't lose any information about the dynamical behavior, but only have to deal with strings of symbols from a finite alphabet and operations on those strings, instead of continuous numbers and non-linear mappings of those numbers. Computational mechanics adds on another layer, by trying to characterize the symbolic dynamics even more compactly, and in information-processing terms. First of all, it gives you a nice, compact description of your system. Second, it constrains any theories you come up with of the causal mechanisms at work, since they've got to be able to match your reconstructed machine, or one or the other is wrong.