To summarize Feynman's explanations in a compact form (which helps to avoid confusion and was discovered by Minkowski in 1908) you just note that charge and current densities together build a Minkowski four-vector field
$$(j^{\mu})=\begin{pmatrix} c \rho \\ \vec{j} \end{pmatrix},$$
which transforms as such under Lorentz transformations. Feynman carefully has to derive this from the assumption that electric charge is a scalar and the kinematics of special relativity, since he has chosen a conventional approach to teach electrodynamics (rightfully so for a freshman course in physics, as which the Feynman Lectures have been invented to begin with, but they turn out to be rather advanced and thus rather make up marvelous theory books for advanced undergraduates to beginning graduates). Of course, a really modern approach, which starts from relativity and formulates electrodynamics as a relativistic field theory, which it indeed is since Maxwell formulated it, is much simpler conceptually (see Landau, Lifshitz vol. II).
In the same way the scalar and the vector potential together (in Lorenz gauge!) build a four vector field,
$$(A^{\mu})=\begin{pmatrix} \Phi \\ \vec{A} \end{pmatrix},$$
and thus the (gauge invariant!) field-strength tensor (or to honour Faraday also known as Faraday tensor)
$$F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$$
an antisymmetric 2nd-rank four tensor. It's components consist of the electric and magnetic field components, because
$$F_{0 j}=\frac{1}{c} \partial A_{j} - \partial_j \Phi=-\frac{1}{c} \partial_t A^{j} -\partial_j \Phi=E^j$$
and
$$F_{jk} = \partial_j A_k - \partial_k A_j = - (\partial_j A^k - \partial_k A^j)=-\epsilon^{jkl} B^l.$$
The latin indices run over ##\{1,2,3 \}##.
In macroscopic electrodynamics there's a corresponding antisymmetric tensor combining ##\vec{D}## and ##\vec{H}##, and the constitutive equations can be formulated in manifest covariant way too.
The so very easily to derive transformations of the field components and the charge-current vector shows you that, as special applications,
(a) a current conducting wire, which is electrically neutral in its rest frame becomes charged from the point of view of an observer moving relative to it.
(b) a pure electrostatic field also has magnetic components from the point of view of an observer moving relative to the corresponding charge distribution, which is entirely at rest in the frame, where you only have electric field components. Of course, from the moving observer's point of view there's also a convection-current density ##\rho \vec{v}##.
(c) a moving uncharged permanent magnet, which has only magnetostatic field components in its rest frame, also has electric field components from the point of view of an observer moving with respect to the permanent magnet.
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