What problem is the Many-Worlds Interpretation actually solving? Is it a reframing of the measurement problem? Before I state my question I want to say I am in no way an expert/professional in this field. I read quite a bit on the subject and I consider myself familiar with the basic concepts but I really want to hear an opinion from anyone who is more involved in that subject.
I have trouble fully understanding the fundamental difference between the Copenhagen Interpretation and the Many Worlds Interpretation of quantum physics in regard to the measurement problem, perceived reality and conscious observers. To me it seems like they are both just different framings of the same problems.
CI tells us, at the moment of measurement of a quantum bit in superposition, the wave function collapses and the probability of one of the possible outcomes becomes 1. MWI on the otherhand states the wave function never collapses but instead at the moment of measurement or decoherence, the system the qubit interacted with enters a superposition too and both possible realities exist perpendicular to each other (branching). However in each of these perpendicular realities the probability for one of the possible measurements just jumped to 1.
So the question of Why does a measurement cause collapse of the wave function? (CI) becomes Why only one of the perpendicular realities can be observed? (MWI).
If I was ignorant and to say 'I don't care if perpendicular unobservable realities exists.' both seem very semilar to me and come down to the same questions:
What exactly IS measurement?, Why can we only observe one of the possible realities? and What is the mechanism to decide which one we observe? (If there is any other than fundamental randomness)
So to all of you more familiar with the subject my question is: What problem is solved by the MWI (other than more elegant mathematics)? And what is fundamentally different between CI and MWI in regard to measurement, observed reality or even conscious experience?
 A: 
So the question of Why does a measurement cause collapse of the wave function? (CI) becomes Why only one of the perpendicular realities can be observed? (MWI).

It's not true that only one reality is observed in MWI. They're all observed, just not by the same people.
I think the question "Why only one of the perpendicular realities can be observed?" can be interpreted in two ways. It could be taken as a special case of the philosophical question "why am I me and not somebody else?" While I don't have a good answer to this question, it's not specific to MWI. The people in the other worlds aren't you (though some of them are very similar to you) and they have the same status with respect to this question as people in this world who aren't you.
One of the most common objections to MWI is that it can't explain why you're more likely to be a person who sees a more likely measurement outcome than a person who sees a less likely outcome. Again, this is a real problem, but not specific to the MWI. There is something rotten about all forms of anthropic reasoning, where you try to figure out the "probability that you're you", even when the other people you might be (but aren't) are just other people on Earth or intelligent aliens in another galaxy. Just like the problem of consciousness itself, no one knows how to solve this, but it can't be avoided (unless you give up and become a solipsist).
Alternately, you may be asking why people only experience certain outcomes, corresponding to eigenvectors of some Hermitian operator, and no one (in any world) experiences outcomes other than those. I can't answer that either without a physical model of conscious experience, but I want to make the point that there's no reason to expect it not to be that way. People often seem to expect distributivity or linearity to apply where it doesn't. In classical logic, $B(P\vee Q)$ is not equivalent to $BP\vee BQ$, where $B$ means belief. (I believe the number of hairs on my head is either odd or even, but I don't believe it's odd and I don't believe it's even.) In discussions of Schrödinger's cat, I often hear the word "ghostly", as though a photo of $\left( α \left|\text{live cat}\right\rangle + β \left|\text{dead cat}\right\rangle \right)$ should look like a pixelwise blend of a photo of a live cat and a dead cat. It just doesn't work that way. Whatever physical state equates to conscious experience doesn't work that way either, evidently.

And what is fundamentally different between CI and MWI in regard to measurement, observed reality or even conscious experience?

What MWI does that the collapse picture* doesn't do is attempt to be a complete model of reality. The collapse picture describes what happens to the wave function when a measurement corresponding to a particular Hermitian operator occurs, but it says nothing about when that process should be applied or which Hermitian operator should be used. It isn't clear in the collapse picture if measurements are "triggered" by something about the state of the world just prior to the measurement, or even what the state of the world is (it's not the wave function). MWI, in contrast, says that the wave function is the complete state of the world, that the times and nature of measurements are encoded in it, and that measurements occur "automatically" in the course of Hamiltonian evolution. Whatever you think of the viability of that model, at least it's a model.

* This is what you called the Copenhagen interpretation, but I think that's a bad name for it: see this answer.
A: The answers posted earlier explain the motivation for considering the Many-Worlds Interpretation (MWI). The motivation can be summarized like this: The collapse interpretation is awful, and MWI doesn't have collapse, so MWI is better.
However, rejecting collapse is not the same as endorsing MWI, and rejecting MWI is not the same as endorsing collapse. Rejecting one interpretation is not the same as endorsing another one. Quantum theory can be formulated and used without committing to any "interpretation" at all, while fully preserving its ability to account for the known experimental facts.
In this answer, interpretation means any idea that can be classified as either $\psi$-ontic or $\psi$-epistemic. See this answer for the definitions. Briefly: $\psi$-ontic means that a given reality can only be consistent with one pure state, and $\psi$-epistemic means that a given reality can be consistent with two or more pure states.
Both collapse and MWI are $\psi$-ontic, but that's beside the point. The point is that we can formulate and use quantum theory without committing to any interpretation. Of course, if we don't commit to any interpretation, then the theory is manifestly incomplete. But it's incomplete in a way that experiments are currently unable to explore, and living with that kind of incompleteness is just business as usual in science. People might fear that not embracing one specific interpretation can make their presumptuous peers think that they're secretly favoring some other specific interpretation, but that's just a sociological problem. If a presumptuous peer can't understand the gaping difference between "quantum theory is incomplete" and "I love pilot-wave theory," that's their loss.
Still, I do think MWI helps with one practical problem. Formulating quantum theory in a way that doesn't commit to any interpretation (as defined above) is possible, but expressing things in that way can make them look unfamiliar, and that can deter people from reading it. As a compromise, I often use MWI langauge. It's a relatively familiar, concise, and memorable way to express some of the math. One could argue that using the collapse language achieves the same goal, but there is a difference: questions spawned by using the MWI language tend to be more productive than questions spawned by using the collapse language. Why? Probably because the ambiguity in the collapse idea is blatant (that's why it feels awful), whereas the ambiguity in MWI is more deeply buried (that's why it feels more elegant). By the time we've thought through things carefully enough to find the equivalant ambiguity in MWI, we're that much closer to realizing that experiments have not yet given us any compelling reason to embrace any specific interpretation. Maybe someday they will, but I'm not holding my breath.
A: I’ll try to be brief. The Copenhagen interpretation can basically be reduced to two axioms:

*

*Most of the time, quantum states evolve in time according to the Schrodinger equation.

*When measurements occur, the state collapses onto an eigenstate of the quantum operator that is associated with the classical quantity being measured, with probability given by the Born rule.

Axiom (2) has seemed deeply unphysical ever since it was first proposed.  To be fair, there are other things about QM (such as entanglement) that were probably even more objectionable to many physicists during the development of quantum mechanics. But axiom (2) seems ridiculous in the context of other physical theories. It is not deterministic; it involves the observer in the definition of reality itself; and the concept of a measurement is quite arbitrary, among other issues.
MWI is basically the idea that we just abandon axiom (2) and we still get the same observable physics. When Everett first proposed it, the idea was murky and almost as magical as axiom (2).  But in the modern context of quantum decohrerence, there is nothing fishy at all.
Basically, the idea is that when you set up anything we might call a “measurement,” what is really happening is a series of entanglements. First, the quantum system becomes entangled with your macroscopic measuring device. Next, you look at the readout on the device, and your body and brain become entangled with both it and the original quantum system. Then you mention the results to your colleague, and he/she also becomes entangled. All of this happens through time evolution of the Schrodinger equation alone. (Look into decoherence for the mechanics of how this happens.) But the different parts of this wavefunction, corresponding to different measurements and different states of your brain and body, have decohered, so they effectively (from our human perspective) act like “parallel universes” that do not interact.
In this picture, there is one universal wavefunction, and its time evolution is completely deterministic.  To MWI supporters, this fits “the character of physical law” (to use an unrelated phrase of Feynman’s) much more than axiom (2).  And you get the same physics, as far as any current experiment can show.
But keep in mind that there are understandable reasons why detractors regard it as pseudoscience:

*

*As far as we know now, there are other interpretations that produce the same physics.

*MWI requires the existence of all of the other terms in the universal wavefunction with other copies of ourselves (i.e., the “parallel universe”). It can rightfully be claimed that in this sense, Copenhagen is simpler; it has an extra unphysical axiom, but only one observable “reality.”

A: The many worlds interpretation drops the collapse postulate from CI. This is good because CI does not quantitatively specify when exactly collapse happens. This means that, under CI, quantum mechanics is an INCOMPLETE physical theory. Because CI leaves us with an incomplete physical theory it does not solve the measurement problem.
MWI drops the collapse postulate and leaves us with unitary evolution of the wavefunction. This leaves us with a complete physical theory.
However, MWI does not solve the measurement problem either. This is because MWI does not explain the relationship between the physical state of the universe (described by the wavefunction) and our subjective experience. "Why do we only experience one situation when our physical brain is in a superposition of having experienced both?".
As an illustrative counterpoint which will explain my paragraph above better and also motivate the entire measurement problem.
CI does*, under certain philosophical assumptions, explain the between the physical state of the universe and our subjective experience. It does this by ensuring humans brains are never in superpositions. This means those brains are in well-defined states. If we assume that mental states are 1:1 with physical states then, bam, we have a correspondence between the physical theory and our experience. This is the goal of any scientific theory.
In fact, this is the entire motivation for CI. Prior to CI we in fact had only bare quantum mechanics, unitary evolution. This is exactly what MWI is! The early physicists thinking about quantum mechanics understood that under unitary evolution we could have situations where macroscopic objects (like cats or human brains) could be in superpositions. If the brain is in a superposition of physical states, how do we explain the fact we only subjectively experience single outcomes? This would require a many-to-one mapping between mental states and physical states. This is philosophically uncomfortable so the CI literally, in an ad-hoc way, just says "no, this is not acceptable" and demands that the wavefunction collapses before this many-to-one mapping becomes necessary.
Note that above I am describing a framework for the mind which assumes a dualist framework and which assumes epiphenomenalism, which is the idea that mental states are "caused" by physical states but not vice-versa. Physicists and scientists in general find talking about this philosophical stuff to be taboo which I find sad because I think this mind-body correspondence is actually central to the measurement problem and both physics and philosophy of the mind could benefit greatly from having a close look. Also, I think that most physicists probably implicitly assume an epiphenomanalistic view of the mind.
To summarize: MWI fixes the physical incompleteness of CI. CI fixes the correspondence between physical and mental states which is not addressed by MWI. Neither provides a fully satisfactory physical theory.
Check out "Three Measurement Problems" by Tim Maudlin. Not sure if there's open access version.
Also check out lots of stuff by Jeffry Barrett. The Quantum Mechanics of Minds and Worlds is good, along with the video on his webpage.
A: (Answers so far are a bit different than what I think is techinically the right answer to the question. Historically, I do not believe the reduction of an axiom of CI was not why MWI was devised -- although I guess this is the main appeal of a lot of people who believe in it.)
Q: "What problem is the Many-Worlds Interpretation actually solving?"A: The Wigner's friend paradox.
Everett was a student of Wigner, and Everett came up with MWI (called relative-state at the time) as a solution to Wigner's friend paradox. In the Wigner's friend paradox, it can be shown that there is a disagreement between the observations of observer making an measurement, and a second observer who is isolated from the first observer. Wigner's solution was that collapse must happen within the mind of the observer, while Everett's solution was the MWI.
A: QM is standardly presented in two sets of axioms. The first set gives a description of the quantum state which evolves deterministically. The second set, describes what occurs on measurement of the quantum state. It collapses to an eigenstate of the observable. This is also an evolution, but unlike the first case, this evolution is not deterministic - it is non-deterministic.
Some physicists do not like this division into a kind of punctuated evolution where a state evolves deterministically, only to collapse indeterministically on measurement. It's worth notimg that in Rovelli's relational QM, all interactions are seen as measurements. Personally, I agree with this.
Many Worlds try to solve this by simply denying the collapse postulate. Personally, I think they are trying to solve a non-problem. There is no philosophical reason why nature should be deterministic in all its aspects. In fact, this question was raised early on with Aristotle recalling in his Metaphysics that some philosophers 'declare that chance is a cause'. Moreover, I don't think it has been emphasised enough that the collapse postulate also gives us a direction of time. It's notorious that classical physics, in the small, is agnostic about the direction of time. For example, Rudolf Haag writes in his book, Local Quantum Physics:

Replacing the word, 'measuring result' by 'event', it is clear with a high level of confidence, a flash on a scintillation screen, the blackening of a grain on a photographic plate, the death of a cat ... can be considered as facts irrespective of the presence of an observer. But can one isolate events with absolute precision? Can one describe the universe as a set of events increading in time? This would neccessitate the introduction of irreversibility on a fundamental scale and a revision of the concepts of space and time ...

A: It's pity that hidden variables aren't mentioned. This isn't an interpretation of quantum mechanics but a whole new theory of some weird deterministic stuff behind the dynamics of the probabilities present in quantum theory. Obviously, the weirdness of the deterministic stuff is the drawback. You can't observe it though the effects are there. You can't observe it but you can describe the stuff mathematically. It's somehow more satisfying than inherent probability. At least, to me (and of course, to proponents of the stuff). So let write some words on this theory.
Louis-Victor-Pierre-Raymond de Broglie initially proposed the pilot wave, which is a physical wave corresponding to the wave function, and David Bohm, later on, devoted much of his work to the construction of a consistent (whatever that means) theory including such a wave. I think that Einstein would have liked the theory, as he couldn't stand the thought that god plays dies. If this direction of thought was given more attention at the beginning of the quantum mechanics tale then maybe that would be the prevalent picture today. Instead of the Copenhagen interpretation.
What does it entail? In the HV interpretation, the wave function is not just a mathematical entity (construct, object, device) to describe physical observations and calculate the probabilities to find these. The wave function is built out of hidden variables in a way reminiscent of the way that a gas or liquid influences the motion of a small particle (Brownian motion) suspended in it. The gas or liquid represents the wavefunction "surrounding" a particle (quantum field theory makes use of Brownian motion too: look here).
This physical wavefunction affects the particle continuously. The particle corresponding to the wave function is corresponding in a literal sense. The particle's position and velocity are well defined at any moment. They are continuously changing though, in accordance with the wave.
What exactly are these variables? Who knows? They are hidden! They are not made out of the same stuff as the stuff out of which the medium surrounding the Brownian particle is made. I'm not sure what Bohm (or van 't Hooft, a Dutch physicist well known in the realm of quantum field theory and advocate of hidden variables) has (had) in mind if one can ever imagine this.
The existence of local hidden variables was ruled out by Bell's experiment, which is an experiment involving quantum entanglement. Not surprisingly! After all, quantum entanglement is a non-local phenomenon. When a measurement is made, the physical correlate of the wave function influences a faraway particle (entangled with a particle on which an observation or measurement is made) instantaneously. This only goes to show how weird these hidden variables (particles?) behave. Note that the instantaneous influence on a faraway particle doesn't mean that the speed of light is superseded. No information is transmitted faster than this speed, though you might think that information about one particle is received by the other particle instantaneously. I remember I had a hard time explaining this to a professor of the philosophy of the sciences, who had even studied physics. I took a number of copies of a popular article to college about the 1982 experiment conducted by Alain aspect, another experiment in the entanglement arena (obviously, quantum entanglement is very sexy!). It's tempting to view the two specially separated particles as a whole (Bohm's most notable book is somewhat new-age-like called Wholeness and Implicate Order.
The hidden variables constitute the wave function. The collections of hidden variables (wavefunctions) collapse if they interact (by one of the spin-1 fields) with other collections of hidden variables. Here the comparison with Brownian particles can't be made, obviously. all the deterministic stuff can simultaneously collapse. One can even argue that spacetime itself constitutes hidden variables surrounding particles, though the concept will always be problematic due to the hidden quality.
One last thing. It was once said that the very act of observing makes the wave function collapse. If this were the case, then life never could have evolved. Everything in existence (before the arrival of humans) would have been in an evolving superposition, meaning that it would be impossible for whatever to evolve.
