The answer is relatively simple, but complicated.
We cannot prove that Peano axioms (PA) is a consistent theory from the axioms of PA. We can prove the consistency from stronger theories, e.g. the Zermelo-Fraenkel (ZF) set theory. Well, we could prove that PA is consistent from PA itself if it was inconsistent to begin with, but that's hardly helpful.
This leads us to a point discussed on this site before. There is a certain point in mathematical research that you stop asking yourself whether foundational theory is consistent, and you just assume that they are.
If you accept ZF as your foundation you can prove that PA is consistent, but you cannot prove that ZF itself is consistent (unless, again, it is inconsistent to begin with); if you want a stronger theory for foundation, (e.g., ZF+Inaccessible cardinal), then you can prove ZF is consistent, but you cannot prove that the stronger theory is consistent (unless... inconsistent bla bla bla).
However what guides us is an informal notion: we have a good idea what are the natural numbers (or what properties sets should have), and we mostly agree that a PA describes the natural numbers well -- and even if we cannot prove it is consistent, we choose to use it as a basis for other work.
Of course, you can ask yourself, why is it not inconsistent? Well, we don't know. We haven't found the inconsistency and the contradiction yet. Some people claim that they found it, from time to time anyway, but they are often wrong and misunderstand subtle point which they intend to exploit in their proof. This works in our favor, so to speak, because it shows that we cannot find the contradiction in a theory: it might actually be consistent after all.
Alas, much like many of the mysteries of life: this one will remain open for us to believe whether what we hear is true or false, whether the theory is consistent or not.
Some reading material:
- How is a system of axioms different from a system of beliefs?
First off, the distinction you're attempting to between "counting numbers" and "natural numbers" doesn't match how the words is usually used in mathematics. To a mathematician, "the natural numbers" in general means the pre-formal concept you're calling "counting numbers". (And the phrasing "counting numbers" generally isn't used at all). In particular, being a model of this or that formal system doesn't qualify anything as being the natural numbers; only the true Platonic natural numbers are natural numbers.
(That's except for people who have learned enough set theory to know how natural numbers are usually modeled there, but haven't yet gained a sufficiently wide horizon on the matter to know that the model is not the actual thing).
Whether the natural numbers are a human invention or not is something of a philosophical morass that I won't go into -- but the mere fact that they were not created by Peano is not a controversial statement.
Now, since we understand the naturals intuitively, at least to a pretty good extent, what do we need axioms for? Well, most primitively, to see how well we can do by reasoning purely mechanically from axioms. For example, it led to a significant improvement in geometrical reasoning when the ancient Greeks started demanding rigorous proofs from axioms rather than just geometrical intuition; it is natural to wish to know whether something similar could be done for arithmetic.
As it happens, it took many centuries after the Greeks before formal mathematical reasoning had progressed enough that it was thinkable that it might be possible to reason about integer arithmetic without depending on intuition. Peano's axioms were not aimed at telling anything new about the integers -- he was simply one of the first to have a sufficiently well developed notion of formal logic that he could hope to make an axiomatic treatment of the integers work.
The goal of the investigation, then, is not primarily to discover something new about the integers themselves, but to see whether a formal system can be made to support a "mechanical" reasoning about them that is rich and strong to conclude what we already know about them intuitively.
The Peano axioms (and their modern first-order successor, Peano Arithmetic) were somewhat successful in that, but nonetheless the crowning achievement of the whole program turned out to be a negative result: Gödel's Incompleteness Theorem tells us that every reasonable formal system for proving things about the integers will be incomplete: there are truths about the natural numbers that it cannot prove.
Nevertheless, it is still interesting in itself to investigate the strength and properties of various such formal systems.
Also, of course, it is pragmatically useful that when something can be proved in a formal system with few axioms to it, then it is very certain to be true about the actual naturals ... there's a much smaller risk of having overlooked something during the proof than if we base our reasoning purely on an intuitive understanding of how the naturals ought to behave. In this way it's kind of a gold standard for a proof that it can (in principle) be reduced to Peano Arithmetic or a similarly bare-bones system. We recognize that there will be things that can't, but these exceptions will be subject to a lot more scrutiny before one considers oneself convinced that they are in fact true.
Best Answer
Peano axioms come to model the natural numbers, and their most important property: the fact we can use induction on the natural numbers. This has nothing to do with set theory. Equally one can talk about the axioms of a real-closed field, or a vector space.
Axioms are given to give a definition for a mathematical object. It is a basic setting from which we can prove certain propositions.
As it turns out, however, it is possible to use the natural numbers as a basis for some of our mathematics, and we can use Peano axioms to model first-order logic (its syntax and the inference rules), and the notion of a proof.
This can be seen as a basis for some of the mathematics we do, however it is often a syntactical basis only: we only use the integers to manipulate strings in our language and sequences of these strings. We do not have the notion of a structure, of a model.
But it seems that you are mainly confused by the use of the term "axioms". These are just basic properties for a mathematical object. In this case "the natural numbers". Much like there are axioms in geometry, but geometry doesn't usually serve as a basis for many parts in mathematics.