I also unfortunately don't really understand why people care so much about them, although I should probably go back and read old TWFs. 

Sort of a combinatorial example of the fundamental groupoid is the category assigned to a graph where the objects are vertices and the morphisms are directed paths, and v->w->v = id_v. (If this makes sense.) I believe that this category is nice because (IIRC) you can read off the definition of graph homomorphism from it.