I think it all comes down to the finite speed of light, or more precisely, the finite speed of the propagation of changes in the electromagnetic field.
Suppose you have a charged particle moving with uniform velocity. Maxwell's equations tell us that electromagnetic waves propagate at a finite speed. As a corollary, we have that the field at a point is not due to the present position of the moving charge, but where the charge was. As I understand it, this is what gives rise to magnetic effects.
We refer to "where the charge was" as the Retarded potential and when it was there as the Retarded Time. If we let x and t be coordinates where we are measuring the field and $x'(t_r)$ and $t_r$ be the retarded position and time respectively, then we have $c|t-t_r|=|x-x'(t_r)|$.
The potential for a static charge is $V=-q/(4\pi\epsilon_0|x-x'|)$. Adjusted for case of a charge in uniform motion it is $V=-q/(4\pi\epsilon_0|x-x'(t_r)|p)$ where $p$ is the magnitude of the vector projection of the vector pointing from field point to charge at time $t$ projected onto the vector pointing from the current position of the field point to the position of the charge at retarded time.
So we have two adjustments of the potential from the static case, a reference to a past position of the charge instead of just present position, and a scaling factor also related to past and present position relationships.
What's the actual physical meaning of the adjustments? Given finite propagation speed of field changes, It makes intuitive sense that instead of the usual distance from field point to present charge location, we'd have to use the distance to some location where the charge was in the past. This only explains one adjustment to the potential formula, but this doesn't explain that factor of $p$.
Is p some scaling factor thrown in just to adjust for relativistic effects or does it correspond to some property of the motion? I have a few guesses.
a) Charge density and current density are part of a Lorentz 4-vector. While the charge is the same in both reference frames, the charge density Lorentz Transforms. So $p$ is just a way of taking this into account. Electric potential is not due to total charge, but how it is distributed. This seems more a mathematical explanation than physical though.
Suppose we had a non-moving point charge, but whose charge increased with time. Then our formula for the potential would be the usual one, the distance factor would be constant, but we'd just adjust the q value according to retarded time.
b) Suppose you have a bunch of identical particles located a uniform distance from each other along a straight line. For a given position $\vec{x}$, consider the field due to these static charges. Consider the amount of field contributed by each charge independently. Each charge contributes a field due to a different distance and a different position with respect to the charge. As a parameterized function along the line, the direction of the Electric field turns a bit. The strength of the field drops with the inverse square of charge distance. The distance to a field point projected in the direction perpendicular to the line is a constant for all points. The one parallel changes.
So comparing fields associated with distinct static charges we already see a sort of "turning" in the field direction if the charges lie along a straight line. We also have variation in field strength. There is also something special about distances parallel vs. perpendicular to the line along which the charges are located. All in terms of electro statics.
If to each charge we associated a function that would start form 0, increase charge at the point, then decrease back to zero in a specific time interval we might be able to break up field changes due to motion between time variation of charge at a static location vs those field changes due to position to get a better, more purely physical understanding of what's going on.
