It's basically because you don't have infinite storage -- I could, given enough time, feed your PC a string that it wouldn't be able to tell was unbalanced. Of course, to do that I would have to overflow your available storage which would take a while.

For all practical purposes, your PC adequately models a Turing machine. In point of literal fact, it doesn't because it only has finite storage. In practical usage, it is sometimes helpful to consider various states your code may enter and how they interact. For understanding how the system actually works, the author appears to have found the state-switching model to be a useful one, and indeed I understand that it's common to model the internal state of various devices using finite automata.