Why do accelerating electrons emit photons?

My friend Alice Sheppard, known on Twitter and elsewhere as @PenguinGalaxy, asked her physics pals to explain why it is that an electron emits photons when it is accelerating/change direction?

There were several replies that suggested this has been thought about a lot but nobody could come up with a simple, solid explanation. There was a bit of hand-waving and a lot of obscure words that I only vaguely rememberd the meanings of. As I understand it, even the great Dick Feynman got it wrong in one of his famous lectures.

Now, I am a lowly chemist, with aspirations, as you all know, to being some kind of award-winning photographer by day and a rockstar by night, and the farthest I got with physics was to successfully pass the first year undergraduate university course after two fails. But, I did try hard, did a lot of background reading and as you also know I have now spent more than thirty years writing about all kinds of science, including a lot of physics and quantum mechanics. So, I had a go at an explanation. Please feel free to pick holes in it and explain its fatal flaws as an answer to Alice’s question.

Electrons have an electric field. If they're moving they have a magnetic field too. If you give an electron a flick you add energy, like flicking a rope, this produces a wave in the electric and magnetic fields together, a wave in an electromagnetic field is a photo. The electron cannot hold on to it, so it is emitted.

I asked a Prof of Physics if my lay explanation passed muster and thankfully he said yes it does ;-)