One of the most important foundational experiments leading to quantum mechanics was the photoelectric effect. Experimentally, it was found that arbitrarily small intensity light could eject electrons from a material, so long as the frequency of the light was above a threshold value. This is impossible to explain with classical wave theory (since the energy is proportional to the intensity). Einstein explained this phenomenon with the idea that light comes in discrete packets of energy (photons) with energy proportional to their frequency, $E=\hbar \omega$, where $\omega$ is the frequency of the photon. Crucially, it is not possible to divide these particles into smaller amounts; when photons of a given frequency are generated they either have enough energy to eject an electron or they don't, there is no middle ground.
Quantum mechanics is a theoretical framework constructed to reproduce this and other basic experimental facts about reality.
Since Einstein's time, there have been many other phenomenal experimental demonstrations of the correctness of the predictions of quantum mechanics. For example, CCD chips can detect single photons, and never detect half a photon. Photons are always detected with energies in integer multiples of $\hbar \omega$, and never with a some fraction of this amount.
Within the framework of quantum mechanics, (independently of whether you are using the Copenhagen or Many Worlds interpretation or some other interpretation that gives physically equivalent results) your question is tautological. The answer to "Why can't one particle be observed to be at place A and place B simultaneously" is that one particle can only be observed in one place at once. If you want to observe something at two locations simultaneously, you need two (or more) particles. Mathematically, to describe one particle, we can write down a basis of states of the form "particle is at location A" and "particle is at location B", and a general one particle state (the wavefunction) is a superposition of these one particle basis states. To describe a two particle state, we would need a basis for states with two particles; such a basis would include a state with one particle at location A and one particle at location B; another basis state with one particle at A and one particle at C; and so on; and a general two-particle state would be a superposition of these two-particle basis states. Note that, if there were some object, that could be detected at two locations at once, the word "particle" would not be a very good word for whatever it was the theory was trying to describe, since it is behaving like "two things" would in our classical world.
A very important and subtle distinction here is the difference between a state and a wavefunction. A wavefunction is a state for one particle expressed in the position basis. A state is more general and abstract, and can be expressed in any basis and describe any number of particles. In particular, to correctly describe what's happening when you detect one particle at location A, your state can't be described a wavefunction for one particle; you should also include the detector in your state. Immediately after "collapse" [Copenhagen] or "in some branch" [many worlds], the state of the system is described by a combination like "particle at A and detector at A saw something". If you insist on using a wavefunction picture, you have to imagine the wavefunction is a function over a space not just including the position of the particle, but also the possible states of detector. Then, immediately before collapse [Copenhagen] or considering all branches [many worlds], there is a "peak" in the wavefunction near "particle at A and detector A lit up" and a "peak" near "particle at B and detector B lit up" but the wavefunction near "particle at A and detector B lit up" is zero. There's a wonderful paper by Mott that describes this very clearly [1].
(I realize it is a bit of a contradiction for me to say that the answer is both tautological and depends on something very subtle; in my defense I would say that the answer is tautological if you fully understand quantum mechanics, but many students fail to realize the subtle distinction between states and wavefunctions at first, and this can lead to all kinds of confusion)
You might ask, why is quantum mechanics constructed like this? The answer is that it was constructed to reproduce experimental facts like the ones that I gave at the beginning. If we observed that particles (or "things" that we probably wouldn't call particles) could sometimes be observed in two places at once, we would use a theory other than quantum mechanics to explain the experimental facts. But we don't, and quantum mechanics has never failed to correctly predict the results of an experiment in the regimes where it can be applied.
[1] Nevill Mott, "The Wave Mechanics of α-Ray Tracks", Proceedings of the Royal Society (1929) A126, pp. 79-84, doi:10.1098/rspa.1929.0205.