None. There is absolutely no proof of many worlds or the multiverse. RE the god of the gaps. It's much more interesting to do physics rather than speculate about what falls outside the purview of the scientific method.

The “many worlds interpretation of quantum mechanics” is loosely that when you do a quantum coin flip, the universe splits into two universes, one for each result.

The reason for this thought is when you work with quantum mechanics, your system has a state that evolves smoothly, but if you “measure” it, the state suddenly snaps to (a random) one of the possible measurement values (when the coin isn’t being observed, it smoothly evolves, but once you measure it, it suddenly takes on a random value). However, if you expand your quantum description of the system to include your measurement device as well as the quantum “coin”, that sudden “snapping” goes away. Instead your whole system smoothly evolves, and it evolves into a “superposition” of the shared state of the state of the overall system in each of the possible measurement outcomes.

Extending this idea, it would seem that whenever you could describe a situation that acts like a “quantum coin flip”, both results happen, and the universe “splits”.

I really want to emphasize that the practical meaning of these “other worlds” is just that things are a lot “fuzzier” when you zoom in than classical statistics would suggest. Not that there’s another universe where you stayed with your ex or took a different career path or whatever.

Also this is an “interpretation” of quantum mechanics for good reason. It doesn’t really have any physical implications. In particular, it’s not possible to go “interact with” those “other universes”.

Most importantly, there are other “interpretations” of quantum mechanics, like that quantum mechanics is really a rethinking of statistics not of physics.

https://arxiv.org/abs/1301.1069

Effectively there is no current consensus on the issue, though Many Worlds and String Theory are widely regarded outside their adherents as non-falsifiable and therefore not legitimate theories.

Essentially the proponents of the theory have created beautiful math that fit their view, but absolutely nothing in the real world that can show that it is a more valid theory than any of the other theories which have equally elegant math to back them.

[H]ow does one look at atoms and say "My God! There must be many worlds than just our one?"

Electrons. You've seen the model of the atom, right? Cluster of balls in the middle (protons and neutrons) and the electrons are little balls that whizz around like little planets around a Sun?

That model is a simplification of the truth. It turns out that it is impossible to pin down where an electron is and also know what it is doing. And if you know what it's doing (you can see its effects), you'll have no idea where it is.

Where they are has to be measured by probability. "It's bound to this nucleus / taking part in a chemical bond so it's likely to be in this vicinity", is about as close as you can get.

There is literally nothing excluding that electron from temporarily being a billion miles away. That's astronomically unlikely, but it's not impossible.

And by some measurement methods, when you do try to pinpoint where the electron is, it can appear to be in multiple places at once.

This can be interpreted as bleed-through from nearby quantum realms, maybe even other universes, where the electron is in one place per nearby universe. One of those places is ours, but we cannot tell which. And by the time we've made any kind of determination, the electron has moved. They never stop.

Photons - particles of light - also do this. All subatomic particles do this.

The more subatomic particles you have in some combined state (as an atomic nucleus, or even a molecule), the lower the probability is that that bound state can be in multiple places at once, but again, it is not ruled out.

But it does mean that the more bound particles an object is made from, the more definite its position appears to be, which is what we're used to at our human-sized scale.

If you want to go into depth on this, I recommend you look up Sean Carrol talking about the subject - or read his book Something Deeply Hidden, if you're up for it - he's one of the best science communicators I've heard and a strong proponent of many worlds.

But to try to summarize it in very short: the "multiversal" behavior is already baked into quantum mechanics - a particle can be in two places at once, as in the double slit experiment - just at a very small scale. Traditional quantum physics postulates that there's some mechanism by which this behavior is cut off before it reaches the macroscopic scale (wave function collapse). Many Worlds just asks "Do we actually need this postulate? What would it look like if we didn't have it?" And the answer is, it would look like the universe we experience, just with a multiverse along side it.

There isn't any "proof"; in fact, Many Worlds is what's called "unfalsifiable", which means we don't have a way through the scientific method to show Many Worlds to be false.

Also, it's not really

My God! There must be many worlds than just our one?

But more

There are moments in time where one path is taken and not another... but what if all paths are taken, somewhere?

It's not meant to be a valid theory, it's just a possible outcome of having a spacetime continuum; because it's not falsifiable though, it's not worth pursuing right now, only worth keeping in mind in case we come across new evidence to evaluate.

I don't understand how it's any more outlandish than thinking that we can be aware of everything that exists, or that everything exists in a straight line through time, never branching. Maybe it's a lack of understanding on my part, but it seems the sum total of what we have discovered through science, or even through imagination, only illuminates a very small subset of reality. We can only measure with the instruments we can imagine and build, and with our own limited senses. So I wouldn't jump to believe, nor to label unbelievable.

I admit that I might be dumb, but, how does one look at atoms and say "My God! There must be many worlds than just our one?"

Well, we looked at atoms and found out that the only meaningful way to describe them is with quantum mechanics. This is the most precise and possibly best tested physical theory ever developed. And it says that if an atom starts out in state A, it will then naturally evolve into a state A+B.

Now, A and B are mutually exclusive. So what does that mean? One reasonable way to view it is that it is indeed physically in both states A and B as the theory says. That's ultimately what leads to the many worlds interpretation. The atom is both in state A and state B, and the universe accepts both of the different trajectories of reality that leads to.

This view is equivalent to a number of other ways of view things, all of which lead to the same prediction of physical behaviour for now, so essentially you can just pick your favourite.

Multiverse and many worlds interpretation are two different things.

The idea of multiverse is that there are many other universes existing in parallel with ours. Either the universes are created through different big bangs, or maybe the universe is constantly splitting into many other universes. This is mostly science fiction.

MWI is one of many competing ideas to help coming to terms with the counterintuitive nature of quantum physics. A particle can be in many places at once when not observed. Once it’s observed, it chooses to stick in one place. MWI is one interpretation of why this is happening.

If you want to know why it's taken seriously:

https://m.youtube.com/watch?v=kTXTPe3wahc

Tl;dr: you need to actually understand the physics at play that lead to serious consideration of the many worlds theory. It's not the pop-sci it gets painted as. It's much more specific.