In my book (Hodel's Intro To Mathematical Logic), we are given several examples of formalized mathematical theories such as group theory, Peano arithmetic, etc. But I've had this ongoing confusion: the formal system can generate theorems, but then the book goes on to talk about particular interpretations that we can provide for the symbols of the language. For example, we are given the standard interpretation N of PA for Peano Arithmetic where we're given a domain of natural numbers, a 0 interpreted as the number zero of the natural numbers, etc.
My confusion is that numbers don't exist 'out there' like tangible things do. It's easy for me to see how a sentence Pa is true in an interpretation where a=John and P='is a person', but not so much for numbers.
So: What does the book mean when we provide the standard interpretation (i.e. what is it that we are interpreting)? What do we mean when we speak of "truths of arithmetic" that lies outside of the formal theory of Peano Arithmetic? Are the numbers (and their operations) defined outside of the system or something?
p.s. this is especially confusing considering I would call myself something like a fictionalist, or formalist, or something of that nature.