The water below is flowing and the rain is falling. Nothing in the image is changing — not one pixel is being rewritten. Only the meaning of eight numbers is.
Every pixel here stores a number, not a colour — an index into a table of 32 colours. To animate the waterfall, the program rotates eight entries within that table. Every pixel holding one of those numbers changes at once, because the number now means something else.
The rain is always in the image. Press make it rain and no pixels are added — those pixels were there the whole time, painted in an index whose colour happened to match the sky. Changing that one entry from sky-blue to white makes them visible. Mark Ferrari built whole days this way: "the same piece of art 'shifted' to different palettes", cycling seamlessly "through the 24 hour light cycle, and even change weather conditions."
And here is the part I find genuinely sad. On 1990s hardware this was free — the video card held the palette, and rewriting eight entries changed the whole screen with no per-pixel work at all. That hardware layer is gone. To show you this in a browser, I have to walk all 40,960 pixels every frame and write each one by hand. The technique survived. The thing that made it cheap did not.