I know that the Maxwell equations are usually the explanation for all electromagnetic phenomena, but I would like to know why those are valid, if there is any reason for them.
Maxwell's equations can be derived from the following considerations:
The theory must respect the principle of relativity (no inertial frame is distinguished from another).
It is a theory involving continuous entities called fields, with matter-like entities called charges, and it is as simple as possible.
The charge property is invariant between inertial frames.
The theory involves a force which does not change the rest mass of the things on which the force acts.
Within these constraints there is little room for manoeuvre left, and Maxwell's equations are pretty much the simplest possibility you can find.
We respect (1) by using 4-vector and tensor notation throughout.
We respect (2) by not introducing things like non-linearity, and preferring tensors of low rank.
We respect (3) be asserting it.
We respect (4) by using an antisymmetric tensor for the field tensor, and making the force from the product of this with a 4-current.
Another way to proceed is to use a Lagrangian method, and again Maxwell's equations are pretty much the simplest field equations that respect items 1-4. (I say "pretty much" because there is not always a completely clear way to determine which theory is simpler than another.)
I assume from the question that you may not be familiar with tensor methods, but I hope the above nevertheless gives some flavour of the ideas. But no theory in physics can be derived in full, like a proof; rather we look for elegant and coherent sets of ideas, and then use empirical observations to discover whether the ideas describe the physical world correctly.
They have been experimentally validated. Therefore, we take them to be true until an experiment invalidates them.
Measuring magnetic fields around wires, the validation of the behavior of inductors in circuits, and the successful description of EM waves are just a few experimental validations.