In This One ‘Anomaly’ Is Driving Physicists To Search For Light Dark Matter, what does Ethan Siegel mean by "The fact that there are no stationary, oscillating in-phase electric and magnetic fields led to Special Relativity."
I suspect that the author is alluding to the fact that Maxwell's equations force the oscillations to be out of phase and thus suggest that these electromagnetic waves must travel at the speed of light. If pressed, I couldn't produce proof of this from Maxwell's equations, but the ideas of special relativity arose from a discrepancy between Newton and Maxwell having to do with the constant speed of light, so it seems a logical leap to think that these oscillations being out of phase has something to do with the definition of $c$ or its status as a universal constant.
In general, the ideas of SR arose from the fact that Newton stipulated constant space and time in his works, but Maxwell claimed that $c$ was universally constant. While the contradiction isn't immediately apparent, one can imagine a case in which a person in a moving inertial frame measures the speed of light in that frame, and a separate observer measures the same light from a stationary frame. If they both measure the speed of light to be different then Maxwell would be wrong, and so would all of electromagnetism. If they measure the same speed of light than Newton's constant time/space must be wrong.
In a traveling wave, the magnitude of the magnetic field is proportional to the magnitude of the electric field at each point in space (they can't be equal because they have different units, but they are related by $|E| = c|B|$). This means the nodes and antinodes of the two fields are at the same places, and hence the magnetic and electric fields are "in-phase." See this picture.
However, in a standing wave (like between two mirrors in a cavity), the nodes of magnetic field are at the antinodes of the electric field and vice-versa, so they are "out-of-phase." See this picture.
My guess is that the author is saying that there is no standing wave where the nodes and antinodes of the two fields are aligned. That is, there is no way to make a traveling wave stationary, even relative to you when you try to catch up to it.
This relates to Einstein's thought experiment that PM 2Ring linked in the comments.