The fact that all laws are equal in all inertial frames of references comes from special relativity being flat. What this means is that Einstein didn't discover the curvature of spacetime yet, thus the only metric valid is the minkowskian one:
$
\eta_{\mu\nu} = \begin{pmatrix}
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
$
As we can only use this metric, we must also refrain from transforming into frames of reference that change its components: we can only use the Lorentz transformations. Why is this? I guess you probably know that:
$
\Lambda \eta \Lambda^{T} = \eta
$
defines a Lorentz transformation $\Lambda$, and so we have that the components of the metric don't change under a transformation. As Lorentz transformations only allow inertiall frames of reference, Einstein concluded that the laws of physics are equal only in these reference frames.
To clarify your doubt about Maxwell equations, what we percieve as a magnetic field is no more than an electric field in a non-inertial frame of reference (and viceversa). So if we accelerate, one of both fields disappears? That is exactly what this means, and you can think about the following.
An electric field accelerates a charge, while a magnetic one rotates it. But if our (non-inertial) frame of reference spiraled together with the charge, we would not percieve the rotation caused by the magnetic field, but only notice the acceleration of the electric one. Therefore, in non-inertial frames of reference one of both fields may dissapear, and this results in Maxwell's laws changing its forms (as the fields themselves would be different).
