I have been low-level systems coder for about 10 years, and have no clue how demos are made besides the fact they use the technique of procedural generation and/or shader(s).
- learn how to program a computing device with some kind of visual (and audible too, ideally) output
- make the coolest looking (and sounding) thing you can get working
- show it off to other people, look at what they made, get inspired to do better
- return to step 2.
There are more details, of course - a lot of the "classic" demo effects depend on writing to a linear framebuffer, which isn't exactly how modern video APIs work - and there's all kinds of tricks that are only relevant to the particular hardware they were invented on - it's not impressive to break the C64's sprite limit on hardware that can draw a billion triangles a second.
My recommendation: find a way to draw pixels to a linear framebuffer (an HTML Canvas element, an SDL1.x Surface, even an OpenGL or Direct3D texture that you can render to a rectangle, if you can handle the complexity of those APIs), then go watch YouTube videos of classic 2D demos like Future Crew's "Unreal" and "Second Reality", and try to replicate those effects in your own framebuffer.
You may succeed, you may invent your own effects that you think look just as good. Once you've got some things that look good, you might want to try optimizing for code-size or speed or mixing two effects together in a surprising way.
And like any creative endeavour, the most important rule is: have fun.
Is there a guide on how to start making one?