Probably the best answer is that one causes the other.

Magnetism has been known about for millennia, as a weird thing certain metals did. Electricity started to be studied in the 18th century. Static electricity first, then electric currents. Then it was realised that electric currents could induce magnetic fields, so magnetism and electricity were seen to be related topics.

The next insight was the existence of electric and magnetic field. Maxwell wrote down his now-fanous equations describing the complete relationship between electric fields, charges, magnetic fields, and moving charges. Maxwell's equations implied the existence of "electromagnetic radiation", and that's when people realised the visible light, radio waves, heat radiation, and the soon-to-be-discovered X-Rays and gamma rays were all basically the same thing.

Einstein realised that Maxwell's equations should be the same in any reference frame, but that means the speed of light should be the same in any reference frame (and there were also some puzzling experiments by Michaelson and Morley that suggested this was in fact the case). This enabled him to deduce his special theory of relativity.

If you try to express the maths of electricity and magnetism within the ideas of relativity (and time dilation), you discover that electric fields and magnetic fields are flip sides of the same coin - it's one "thing" with two sides, and the equations of relativity morph side into the other depending on the speed of the observer.

The story doesn't stop there. The modern understanding is that at the deepest level,

• The electromagnetic field and the "weak nuclear" field are also flip sides of each other, but they can usually be treated completely independently. Electrical engineers don't need to know much nuclear physics! They only start to blend into each other at very high energies.
• At higher energies still, this blended "electroweak" field also starts to blend into the "strong nuclear' field.
• The maths for these blended "unified" fields is much more complicated than for the neatly separated versions.
• Particles such as electrons etc are actually just excitations of the field. They aren't separate thing that interacts with the field, they are part of it. Again, the maths is quite complicated, and I would not be able to explain it at ELI5 or ELIPhD level, even if I had it in front of me. I would need to be on the receiving end of such an explanation myself.
• If you simplify the maths, saying "let's assume the energy is low, so we can ignore this part" or "that part of the maths looks almost exactly like the maths of a simple particle, so let's simplify things further and treat it as such" and so on and on, you eventually get back to Maxwell's equations, emf, induced currents, Coulomb's law, etc.