In other words, why should I learn about it? When am I going to say... oh I need to know about push down automata or turing machines for this.
I am not able to see the applications of the material. Thanks
There are problems that are nice fit to this kind of solutions, some of which are:
There are probably many others. If you start writing code that has some ad-hoc state variable depending on which some functions can do this or that, you can probably benefit from proper FSA.