I like Muphrid's explanation #9 a lot. Let me see if I can explain it a little. In one picture of relativity, nothing is at "rest" because everything is moving forward in time, which is treated as a coordinate much like space. Something moving in space is actually moving in some direction in space-time. An acceleration is a (hyperbolic) rotation in space-time.
In Newtonian physics, the physics of a system does not depend on what angle you look at the system in. If you rotate the system, calculate the equations of motion, and rotate them back, it shouldn't change the results. The same is true for shifting the overall velocity of the system and shifting it back. This is called Galilean invariance and predates Einstein, but relativity is needed to make this compatible with the idea of a constant speed of light. In relativity, the shift is velocity is actually a hyperbolic rotation, which unifies the Galilean invariance into a rotation invariance.
The trajectory of charged bodies can't change if you rotate the system and rotate it back! Consider a wire containing equal amounts of positive and negative charges, with a net current. For simplicity, imagine the positive charges are at rest and the negative charges are moving along the wire in one direction. What is the force on another charge outside the wire? Well, if we don't know how to calculate the magnetic force directly from the current, we can transform into another frame where the current is zero (where the positive charges and negative charges are moving with opposite velocity). Because of length contraction, the positive charges are closer together and the negative charges are farther apart, so there's a net positive charge. This charge exerts a force on the external charge, which causes it to move. Which means in the original frame with a current, there needs to be a force which causes the external charge to move. But there is no electric force, because the charge in the wire is zero. This is called the magnetic force.
But we can see that the magnetic force is just the same thing as the electric force in a moving frame of reference, so we call it the electromagnetic force.