I saw this 'theory' on a YouTube video - It is my untested theory, that all atoms move relative to one another and it is matter which curls around electro-magnetic radiation, rather than EMR actually moving.
Since current is EMR, the electrons in the wire do not actually move. As far as special relativity, you've got to get your reference frame according to the fact that the wire is on earth, which is spinning; the earth is rotating around the sun; the sun is rotating around the milky way galaxy; and the galaxy is also moving. Taking all these motions relative to the Higgs field, which is the only possible 'unmoving' lattice, the magnetic field and associated electric field are vibrating, and it is the matter in the wire which is moving. So, pinpointing the reference frame is very difficult as matter occupies space which is whizzing by the stationary Higgs field. As far as how special relativity explains magnetism in this question, I would say the answer is not.