I've been considering this question, and found many people asking the same (or something similar) online, but none of the answers seemed to address the core point or at least I wasn't able to make sense of them in that regard. I'm looking for a layman-friendly explanation.
I'll copy the clearest exposition of this question I found, from here:
What is it that causes an electromagnetic wave to oscillate? I.e. what is the medium that constrains it and pulls it back from one side to the other?
Discounting pressure waves such as sound, as they push backwards and forwards through air, other waves, with amplitude, need something to constrain them. For instance waves in water are constrained by surface tension and gravity so at the end of the "up" gravity and surface tension pulling the water flat pull it back down and vice versa at the end of the "down".
So what is doing that with visible light, gamma rays, microwaves etc? What pulls their energy back and forth causing them to oscillate?
One of the answers there claimed "The electric field constrains the magnetic field and vice versa." Is this true? I didn't see anyone else making a similar claim.
The above pretty much sums up my question, but for completeness' sake, I'm also including below references to some related threads I found:
This guy asked "Do photons oscillate?". The responses explained nicely how his conception of photons was incorrect, but didn't clarify his question of why the wave forms.
Somebody asked a pretty similar question on reddit: "Why do photons oscillate?", adding "I don't understand what makes them swing back and forth." -- again, the answer explained that photons aren't really "particles of light", and don't oscillate, but again no satisfactory answer to the "why" question was included.
Finally, this page has a great applet demonstrating the basic principle of how disturbances in one charge propagate to nearby ones. The oscillation is provided by a spring for illustration purposes. The text naturally raises the question: "Yeah, that all makes sense, but don't expect me to believe that particles as small as electrons are attached to springs. How is [the first] electron made to wiggle -- I mean, how is its speed or direction of motion changed?" -- but I couldn't find the answer in the following pages.
EDIT: I emphasized some passages above to make it clear I am not assuming the EM field moves in space (I realize what is changing is the amplitude over time), and that what I'm asking is indeed, to quote Bjorn Wesen's answer, "why does the source amplitude oscillate in the first place".