Because we don't have direct access to reality. Axioms exist as parts of models, which are human abstractions. Thus at some point to formally talk about reality

Because we don't have direct access to reality. Axioms exist as parts of models, which are human abstractions. Thus at some point to formally talk about reality we have to make rigorous assumptions about it. That human beings are as good as we are at finding axioms that appear to correspond pretty well to reality is amazing to me. It's a really interesting philosophical question to ask why it is that we are.