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I have recently learnt about Hubble's Law and this is my understanding of it:

Galaxies move away from us at a rate that is proportional to their distance from us. So, considering an arbitrary unit of distance d, if a galaxy at a distance d from us is moving away from us with a velocity v, then a galaxy at a distance 2d is moving away from us with a velocity 2v. This proves that the universe is expanding.

But the part that I can't quite visualize is the fact that it doesn't matter where you are in the universe. Regardless at what point in space we are standing, it will look like the galaxies are moving away from us. How is this possible? Are the galaxies moving outward? Should I visualize this as the galaxies being on the surface of a sphere whose radius is increasing?

How do I gain intuition about this and better visualize this?

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Think about it as if you were baking bread (or cake, whatever you prefer). When you bake a bread with raisins, it rises in all directions (due to the yeast). Every raisin in a rising loaf of raisin bread will see every other raisin expanding away from it. Now suppose the raisins as galaxies and the yeast as the Hubble constant.

An illustration from Hyperphysics (Georgia State University) to illustrate this further: Expanding universe

When you now suppose you are the raisin at the edge, from your perspective, you are standing still and the other raisins are receding away. It is all about the perspective, the body which you take as a reference (like with relativity).

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Here are some galaxies (in the first line), and the same galaxies a bit later (in the second line) enter image description here

If you are in the red galaxy (the one in the middle), galaxies that used to be one space away are now two spaces away, so they are moving away with a speed of one space per unit time. Galaxies that used to be two spaces away are now four spaces away, so they are moving away with a speed of two spaces per unit time. All of the galaxies seem to be moving away, and the further away they are, the faster they go.

If you are in the purple galaxy (second from the left), the red galaxy (in the middle) is moving away with a speed of one space per unit time. The turquoise galaxy (fourth from the left) is moving away with a speed of two spaces per unit time. So you also see all the galaxies moving away from you, and the further they are, the faster they go.

Whichever galaxy you are in, you see all the other galaxies getting further and further.

I don't know how colorblind friendly this drawing is. If you have a hard time distinguishing two colors, say so in the comments, I'll try to do better.

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another way to think about this is to imagine what some magic observer expanding with the universe would see

they see galaxies as (mostly) fixed in place - with some small individual random motion - but those galaxies, and everything within them, appears to be getting smaller

this frame of reference is what we call the comoving frame and we use it a lot in the mathematics of an expanding universe - like a rotating frame it is arguably aphysical (it introduces extra forces to account for the transformation, and no being following the laws of physics could inhabit it) and thus we rarely talk about observers in it

but, if we did, we'd have an observer who sees a mostly stationary universe in which the physical constants are changing such as to shrink down every bound system - effectively changing the units of measurement

if our magic comoving observer peered into a galaxy and saw a rapidly shrinking physical observer they would agree that the measured distance between galaxies is increasing, but not because they're moving apart - because the meter-long stick used to measure the constant distance keeps shrinking

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