If I'm on a train, I don't actually need to factor that into my calculations if I want to do calculations about what would happen if I throw a ball. I can just treat the carriage as stationary and do standard ballistics. Or, if I like making life complicated, I can treat the carriage as moving at 60mph - and do standard ballistics again. I just add 60 to all the initial velocities and add 60t to the positions at time t, and the maths just works.
This is because the laws of (Newtonian) physics are Galilean invariant. Mathematically, in any of the equations, you can replace ##x## with ##x'=x-vt##, where ##v## is a velocity in the x direction, and it all just works.
Electromagnetism doesn't work like that. You can describe a charge moving past a magnet or a magnet moving past a charge. But you cannot transform one solution into the other using Galilean invariance. Ether theories of increasing complexity were an attempt to fix that, but experiments eventually led Einstein to the realisation that electromagnetism didn't work if you replaced ##x## by ##x-vt##, but did if you replaced ##x## with ##x'=\gamma(x-vt)## and ##t'=\gamma(t-vx/c^2)##, where ##\gamma=1/\sqrt{1-v^2/c^2}## - the Lorentz transforms.
Crucially, he also realized that Newton's laws of physics were very slightly inaccurate. He wrote down laws of physics - conservation of momentum and energy and the like - that were the same under the Lorentz transforms, not the Galilean one. So you can still treat the train as moving or stationary as you like - but just adding 60 to the velocities turns out to get you an ever so slightly wrong answer. You'll never notice at everyday speeds, but the point about electromagnetism - and light in particular - is that changes propagate at around the speed of light. So "accurate enough for everyday speeds" doesn't cut it.
Hope that's helpful...
