How do we prove the existence of a multiverse? How do we prove that a multiverse exists?
Scientists are talking about our universe not being the only universe, but even if that is true, how can we prove the existence of multiverse? We are being 'confined' in this universe and there is no way we can know what is happening outside, right?   
 A: By definition we'll never be able to observe other universes directly, because if we could they'd be in our universe.
However suppose (for example) string/M theory does get developed into a theory that is easily testable, and suppose we find that all the predictions string theory makes about our universe are experimentally found to be true. That means we'd be pretty convinced that string/M theory was the right theory to describe physics. If this hypothetical development of string theory still predicted a multiverse then we'd be inclined to believe in on the grounds the theory gets everything else right.
It's not unprecedented for a theory to predict things we can never observe. For example we cannot see what is behind the event horizon of a black hole (at least, not without jumping in :-). However we believe what General Relativity tells us about the interior of a black hole because it gives the correct results for everything that we can observe. The multiverse idea is a lot more extreme than the interior of a black hole, but you can see how we could be persuaded that it does exist.
I suppose for completeness I should mention that some scientists have suggested different universes may collide, and we could see evidence of the collision. See this Science Daily article for a introduction to this idea. I'm not sure how seriously this idea is taken.
A: You don't.  That's because Multiverse is not a scientific hypothesis.  It speculates that there are numerous universes out there.  Those are not observable.  They would have each their own set of laws of physics much different than our own. As conceptualized, the Multiverse theory can’t be tested.  For this reason, the majority of physics theorists state that Multiverse does not belong to the body of sciences. 
One leading physics theorist, Nima Arkani-Hamed, sees it differently.  And, he even came up with a Higgs boson mass threshold (140 GeV) that, in his view, would support Multiverse.  However, other physicists have rebutted his logic by arguing that given the speculative nature of Multiverse there are no such relevant threshold applicable.  For a threshold to be relevant you would have to experimentally prove that a Higgs boson of 140 GeV does confirm the existence of Multiverse.  That’s not going to happen.    
A: This depends on the structure of the multiverse. In case of quantum mechanics it is possible, at least in principle, to come up with an experimental proof that a Many-World's multiverse exists. The following thought experiment has been devised by David Deutsch. Here one considers measuring the outcome of an experiment where two possible outcomes exist due to the measured system being in a superposition of the two states corresponding to a definite measurement outcome. According to the Many World's Interpretation (MWI), if one of the two outcomes is actually measured, the other possible outcome is also realized in a parallel World.
To prove that this other outcome really exists, one can consider performing the measurement inside a completely isolated system. If indeed the two possible outcomes really exist, then the entire system can be evolved back to the initial state. But, of course, that would erase the memory of the observer and so the observer wouldn't know that any measurement has actually been carried out. However, instead of evolving the state back to the initial state, it is also possible to let it evolve to a state where the measurement outcome is erased but the record of the fact that a measurement was actually made is kept. In principle, all possible so-called unitary transforms can be realized as time evolutions, and erasing the outcome while keeping the record that a measurement was carried out, is a unitary transform.
Now, if the state of the system had collapsed into a single state corresponding to the observer having observed one definite outcome, then carrying out the above transform will have changed the state of the measured system, while in the MWI it will yield the original state. The observer can then do additional measurements on the system to see if the state is the same, this requires repeating the experiment a few times and then carrying out an additional measurement to see if the probabilities match up with what one would expect if the state of the measured system is unchanged.
