Quantum mechanics, like Special Relativity, is today proven to be true by countless real-world technological devices in common use.
These theories were doubted, however, when they were first proposed, because in various ways they contradicted our common assumptions of how the world worked. Over time, people came to accept that the difficulties in accepting Special Relativity, and perhaps even General Relativity, were more illusory than real. But quantum mechanics has proven to be more difficult to fully accept, or to understand in a satisfying manner.
In quantum mechanics, a fundamental particle such as an electron is no longer considered to be a tiny billiard ball with a static electrical charge. Instead, its real nature is as a wave function.
And the behavior of this wave function is described by Schrödinger's Equation. If one determines the position of the particle to a high accuracy, the wave function looks like a short blip in the area where the particle is located; if one determines the velocity of the particle to a high accuracy, the wave function looks like a wave of a single well-defined frequency and wavelength, but spread over a large area.
The Fourier transform of a short blip includes a wide range of frequencies, so a wave function has to extend over a large area to have a well-defined wavelength; this is why the Uncertainty Principle, which limits the accuracy of simultaneous measurements of related complementary observables, is a consequence of the Schrödinger Equation.
Since velocity and position can still both be known at the same time to a limited accuracy, with the necessary uncertainty split between both variables, then large objects can still behave classically? It turns out that this isn't the whole reason why, as quantum mechanics still predicts some forms of strange quantum behavior even for macroscopic objects.
To highlight this problem, Erwin Schrödinger proposed the famous paradox of Schrödinger's Cat; a cat in an isolated chamber will be exposed to a poisonous gas when a radioactive atom decays; but until the isolated chamber is opened, the atom's state isn't forced into what is called an eigenstate of whether it is decayed or not by means of being observed... and so inside the chamber, the wave functions must be in a superposition of the cat being alive and the cat being dead.
Basically, a quantum particle that is in a fuzzy state makes anything it touches fuzzy too... until it gets "observed", with observation being a rigid Procrustean bed that forces it into one state or another. But what counts as observation?
The Copenhagen Interpretation of quantum mechanics said that classical rigidity proceeds outward from conscious minds; an instrument for observing the state of a particle is held in a classical state because one of us is watching it, and so it forces the particle into the kind of state it is designed to observe... instead of being afflicted with that particle's fuzziness.
The example of Schrödinger's Cat was meant to highlight how this interpretation of quantum mechanics is absurd.
One way to deal with this is to switch to the many-worlds interpretation of quantum mechanics. Here, the real world is all quantum, filled with messy superposed wave functions - it's only inside our minds that we percieve only one classical reality at a time.
There is also the pilot wave theory of de Broglie, which proposes an additional wave that has the function of picking which of the possible outcomes of an observation is the one that will take place; this wave has properties compatible with quantum mechanics, and so is not rendered impossible by the proof that "hidden variable" theories can't work.
And this is connected to the famous Einstein-Podolsky-Rosen paradox. When a certain type of radioactive decay sends two particles off in opposite directions with spins that cancel out, no matter which polarization of spin is measured, the two particles are always found to have opposite spins.
But if one measures the spins with slightly different polarizations, the probablity of the spins not being opposite in general direction increases as the square of the difference in polarization. This means the particles can't just know in advance whether they will have spin up or down for any given polarizaton of measurement (the hidden variables case), because then the probability of not being complementary would be larger for small angles, being at the least directly proportional to the difference in polarization.
An even more trivial solution would just to accept that even if cats can't do calculus, they're still conscious and thus count as observers, although that only deals with the specific illustration, and does not address the "real" problem.
I propose another, simpler, way to look at all of this.
Let us assume that the "real" laws of physics include a modification to quantum mechanics that makes it nonlinear, so that large, macroscopic objects, merely by virtue of their size, avoid becoming fuzzy, and are capable of being used to perform the act of observation.
One such theory that has recently been proposed is that of Roger Penrose; since General Relativity claims that gravity is the curvature of space-time, and this is not comptatible with quantization, things become classical when they are large enough to have appreciable gravitational effects.
Then we can make the Copenhagen Interpretation useful despite it being absurd if it were genuinely true.
Since we have, even at present, let alone at the time when the Copenhagen Interpretation was proposed and accepted - growing out of discussions between Werner Heisenberg and Niels Bohr that took place in Copenhagen - no experimental evidence of the nonlinearity that perhaps needs to be added to quantum mechanics, then, to make it work as a description of the real world, in which large objects never partake of quantum fuzziness... the Copenhagen Interpretation would serve as a stopgap theory that does not prejudice the nonlinearity question by favoring one kind or degree of nonlinearity over another!
That, to my mind, settles the question nicely. The Copenhagen Interpretation is useful, but since we were never meant to take it seriously, the fact that it is absurd doesn't matter! So we can all go back to sleep and not worry that quantum mechanics poses any real philosophical problems!