None.

It is, however, actually simpler than other theories, in that if you just let quantum mechanics do it's thing without extra (unknown) parts to limit it, it produces many worlds. So, by Occam's razor...

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.

We exist on the electromagnetic spectrum of energy, there are other types of energy and waves outside of that spectrum that we are aware exist, thus it’s reasonable to think there might be other realities outside of what we experience, this has lead to the idea of multiple universes, but that is a misinterpretation, as if we ever found ourselves outside of the electromagnetic spectrum we would cease to exist, as our matter does not exist in that spectrum. There is no evidence of any sort of butterfly effect multiverse sort of nonsense, that’s woo woo bullshit science fiction.

Anyone referring to the double slit experiment or schrodinger’s cat as proof of a multiverse don’t understand the basic concepts behind both of those things.

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.

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.

[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.