In what sense are math axioms true?

You only need to "prove" an axiom when using it to model a real-world problem. In general, mathematicians just say "these are my assumptions (axioms), this is what I can prove with them" - they often don't care whether it models a real-world problem or not.

When using math to model real-world problems, it's up to you to show that the axioms actually hold. The idea is that, if the axioms are true for the real-world problem, and all the logical steps taken are sound, then the conclusions (theorems etc.) should also be true in your real-world problem.

In your case, I think your example is actually a convincing "proof" that your axiom (commutativity of addition over natural numbers) holds for your real-world problem of counting stones: if I pick up any number of stones in my left- and right-hands, it doesn't matter whether I count the left or right first, I'll get the same result either way. You can verify this experimentally, or use your intuition. As long as you agree that the axioms of the model fit your problem, you should agree with the conclusions as well (assuming you agree with the proofs, of course).

Of course, this is not a proof of the axioms, and it's entirely possible for someone to disagree. In that case, they don't believe that the natural numbers are valid model for counting stones, and they'll have to look for a different model instead.

The problem of what is means for something to be "true" is a general problem in philosophy which has received a lot of attention, so it is impossible for anyone to give a short and complete answer here. The Stanford Encyclopedia has a nice article on truth. Mathematics is a useful test case for philosophers so a lot has been written on mathematical truth.

There is a separate problem that the word "true" is used to mean several things in mathematics: it can mean just "true", or it can mean "true in a particular structure". For example, the latter meaning is intended when we say that the axiom of commutativity is true in some groups and not in others. The notion of truth in a structure is well studied in mathematical logic. But I think the question above is about plain "truth" not about "truth in a structure".

The easiest way to define what plain "truth" means is to believe that there is some "real" mathematical structure, consisting of mathematical objects that actually exist. This viewpoint is called mathematical Platonism or mathematical realism. Then a statement is "true" if it is true when interpreted as a statement about these real mathematical objects. For example, from this viewpoint the statement "Addition of natural numbers is commutative" is true because the actual addition operation on the actual set of natural numbers is commutative.

There are other "anti-realist" theories of truth that do not presuppose that there are independently existing mathematical objects that can be used to test the truth of a statement. (One problem with the realist versions is that it is far from clear how we would be able to tell whether mathematical objects have various properties using our five senses.) Some go as far as replacing truth with provability; for example, this is one way to understand the motivations for intuitionism. But most mathematicians maintain that there is a difference between truth and provability.

There is a separate issue that most of the time the word "true" is used in mathematical proofs, it is just a turn of phrase that could be omitted. For example, instead of saying "We know that $A \to B$ is true, and we have proved $A$, so $B$ is true", we can say "We have assumed $A \to B$, and we have proved $A$, so we may conclude $B$". This shows up when the proofs are formalized: formal proofs in most theories (e.g. the theory of groups, ZFC set theory) do not have any way to refer to "truth", they simply manipulate formulas. The idea, of course, is that if the assumptions are true then the conclusion is true. But the formal proof itself will not make reference to plain "truth".

The question goes on to ask how we would know (in a realist theory, for example) that addition of natural numbers is commutative. Someone could say "you prove it from other postulates" but then the problem would be how to know that those postulates are true. In the end, the question is how to know that any postulate about the actual natural numbers is true. This is a major issue for mathematical realism, as I mentioned above. The most common answer is that humans have some form of insight which allows us to determine the truth of some (but not all) mathematical propositions directly, without having a formal proof of those propositions. The commutativity of natural number addition is one of those propositions: by thinking about addition and natural numbers, we are drawn to conclude that the addition is commutative. In the end this is how we justify all postulates in geometry, set theory, arithmetic, etc. The realist positions is that although we cannot prove them formally, we can come to believe they are true by thinking about the objects they describe.