The significance of state complexity in automata and regular languages?
State complexity is really about concise description of an object (in this case, a regular language), not about computational complexity.  The general topic is called "descriptional complexity" in the literature and draws its inspiration, in part, from the classic 1971 paper of Meyer and Fischer entitled "Economy of Expression by Automata, Grammars, and Formal Systems" (see http://people.csail.mit.edu/meyer/economy-of-description.pdf ).  This is still an active area, with a yearly conference (DCFS - Descriptional Complexity of Formal Systems).

As for applications, any place where your program essentially relies on a finite-state machine (e.g., parsers) it will be good to have this finite-state machine as small as possible.