They’re not axioms: They’re experimental results.
Coulomb, Faraday, etc did a lot of experimental work to observe and systematize the underlying phenomena. Maxwell then reformulated them (though not in the modern form) and added the displacement current term which itself was later confirmed experimentally.
So the “why” historically comes down to “because people observed this”
Today, the underlying “why” is “because these emerge from the quantum field theory of QED” and even fro electroweak interactions. But that’s a very long leap to make intuitive.
So if you want an intuitive understanding of Maxwell’s equations themselves, the best place to look is at the experiments that underlies them. It’s a lot to put in an Answer, because it’s a lot of physics.
For example, forces on small charges led to Coulomb’s law, which led to the idea of an electric potential hence electric field and Gauss’ law.