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Ahhhhh, I think I get Spherical Harmonics! I'll try to explain in simpler words, roughly, assuming I got them correctly (which I am not 100% sure about)... I don't guarantee it'll be ELI5 though, so it may or may not work for you...

Let's start from a single guitar string. If you pluck it, it makes a sound. It's because the string vibrates. In sound processing (a.k.a. "signal processing"), it is said we can express any complex vibration of a string (or, a sound wave) as a sum of increasingly compressed ("higher frequency") sin/cos waves (called "higher harmonics"), each of them multiplied by its "contribution" (some frequencies, a.k.a. harmonics, are more present, others less). (This is also called more generally a "Taylor polynomial" IIRC/IIUC, or a "Fourier transform" in the particular case of a wave.) Notably, a .MP3 file format takes this sum, and cuts it off at some point - assuming that if we keep only a bunch of the "most strong" harmonics, and cut away the remaining waves that are less "contributing", the audible difference won't be noticeable. Also, a guitar string has a very tiny amplitude of those vibrations compared to its length, so they are barely visible. If you take a friend and start waving a piece of rope between you, you can get bigger waves, making the amplitudes much more visible.

Now, a guitar string is a 1-dimensional wave. If we go to 2D, we get a membrane of a drum. When you hit a drum to make a sound, it will start vibrating. In the same way, the shape the membrane takes in those vibrations, can be expressed as a sum of "simpler" 2D vibrations - presumably mathy/physicsy people call them "circular harmonics" or something. Again, on a drum the vibrations aren't really visible to naked eye, but if you instead took a floppy rubber circle loosely stretched on a metal ring, and start shaking it, probably you'd get bigger waves. Interestingly, IIUC, a JPEG image is basically "MP3 but in 2D case".

Now, back to Gaussian Splats and Spherical Harmonics - I assume that "spherical harmonics" are the same thing but done to a balloon. If you pump up a ("perfectly spherical") balloon, and then hit it, presumably the vibrations of its surface can also be expressed as a sum of increasingly more wrinkled ("higher harmonics") sphere-like shapes, each one multiplied by its factor/contribution/strength/presence in the actual vibration. Again, on a balloon the deformations from ideal shape are super small; but if you imagine some really floppy balloon-like sphere floating in the air, you could imagine the wrinkles being much deeper.

I assume in case of gaussian splats, apart from storing factors of each of the spherical harmonics contributing to the final "distorted blobby balloon shape", you also probably store a color of this contribution. This way, from some angles the dominating color of the blobby balloon would look more green, from others more yellow, etc.

Interestingly and coincidentally, a similar thing happens in an atom. The various "contributions" to the "blobby balloon" shape are called "electron energy levels" (or "orbitals") IIRC (https://en.wikipedia.org/wiki/Atomic_orbital). And the actual "blobby balloon" shape is probably called an "electron cloud" IIRC. I'm super grateful you pointed me in the direction of trying to understand Spherical Harmonics, because when I saw those shapes of atomic orbitals in the past, they always seemed weird to me, and confusing. Now it seems I understand where they came from, that's super exciting!

Found a decent video series about this on YT, showing vibrations of a plucked string, etc.: https://youtube.com/playlist?list=PLpBx-1imHuxISNflNHo0Qr4mQ...

Eheh, found one more video - building up the shape of the surface of the Earth from a sum/superimposition of increasing number of Spherical Harmonics - https://youtu.be/dDQTHFeJf5M - again roughly what an MP3 or JPEG algorithm does, depending on how much "fidelity" you choose, i.e. how many more precise harmonics you keep :)

Ah, and also - IIUC, in some other domains of math, those "harmonics" can also be said to be "eigenvalues" (https://en.wikipedia.org/wiki/Eigenvalue), and in somewhat more familiar territory, they could be called "orthogonal" meaning that a sum of them can allow to represent any shape in some space - in a similar way as orthogonal vectors of a cartesian coordinate system (i.e. the "1"s on XY axes in 2D, or on XYZ in 3D - or your green/blue/red arrows in Blender) allow to represent any point in that coordinate system.