I'll skip explaining EM waves, as I'm sure someone else could do it better (my best explanation that I could come up with just ended up being confusing), but as for (V)SWR, impedance, reflection and whatnot, watch this video. It really is excellent.
I thought it was a good analogy. OK, the people are flowing in a resonant cavity, in and out, in and out, with what should be a sine wave flow rate of "electrons", the partial door works but the real problem is this is a resonant cavity, and to maximize the flow, and hence power, you need to think of this flow of electrons in and out of the antenna in this resonant way.
OK I'm not doing any better.
It's MAGIC. If you can explain what a magnetic field is, and what and electric field is, and what an electron is, then you can intuitively understand this. Heck they can't explain the speed of light, or gravity, or why the traveling of these photons of electromagnetic traveling waves are perpetual motion machines. As well all matter that has propagated through space for the last 13+ billion years.
We use Maxwell's equations, Faraday's laws, etc. It just works. Heck one of the posters here has Fourier's transform as his user name, can he see the convolution integral that maps frequency domain to time and back again? I can't, but Laplace and Fourier could.
If anyone can provide an intuitive explanation, I'm all ears.