Can we exlain Bell's Theorem without dwelving into Quantum Mechanics? I have recently started learning Quantum Mechanics. We learnt some things on light polarization, so I referred to this site. Luckily I stumbled upon this answer which links to this site which sated my quest for an answer. However, I recalled a video titled "Bell's Theorem: The Quantum Venn Diagram Paradox" by MinutePhysics. Is the classical answer insufficient? Why is it insufficient?  
 A: The classical answer is indeed insufficient.  The way to attempt a classical description of a Bell experiment would be to imagine the probabilities for experimental outcomes as being given by a probability distribution over some set of "actual, physical states," called ontic states.  The ontic states are not in general experimentally accessible; you can only measure some of their properties, so they are also called "hidden variables" (some people also use this to mean the probability distribution over them.)  Crucially, the ontic states are seen as existing independent of measurement, and measurement is seen as revealing some of their preexisting properties.  This allows us to preserve our classical intuitions of measurement, i.e., that when we measure some quantity, its value already exists before our measurement.
For example, in the "three-polarizer" experiment you linked to, assuming the light gets through the first filter, you could imagine the possible ontic states to be:
$$\{00\},\{01\},\{1x\},$$
where $\{00\}$ means "will make it through the second and third filters," $\{01\}$ means "will make it through the second filter, but not the third," and $\{1x\}$ means "won't make it through the second filter."  Then we can describe the whole experiment classically by just assigning probability 1/4 to each of the first two ontic states and probability 1/2 to the third.  These probabilities are seen as just encoding our lack of knowledge about the ontic state of each photon, rather than something intrinsic to the photon.
The Bell (EPR) experiment is more complicated, so since there are good resources available to describe it (the wikipedia article on Bell's Theorem is fine), I will not reproduce the details here.  But without going into those, what it basically shows is the one can create an experiment where the (probabilistic) predictions of quantum mechanics cannot be reproduced by any probability distribution over any set of ontic states.  Since such experiments have subsequently been performed, and since they confirm the quantum mechanical predictions, we conclude that quantum mechanics cannot be described classically in this way (by a local hidden-variable theory).
