What does adding dark rum give to a fajita stir-fry?
It adds the flavour of rum.

I'd love to go into a detailed exposition of how rum is a fantastic pairing for "latin spices", but as far as I can tell, no such relationship is documented.  It's just a theme ingredient, something that the author evidently felt would make an interesting combination.

Rum is more commonly associated with sweet fruits, chocolate, and apparently certain dairy products.  It's not something I'd consider adding to the typical fajita mix (peppers, onions, cumin, chili) - seems like a waste of perfectly good rum to me.

If you don't want to spend the money on rum, just buy some rum extract in the grocery store.  It's not as good as genuine rum, of course, but as a first-time recipe experiment it will give you a pretty good idea of whether or not it's worth using real rum the next time (assuming you like the recipe at all).