It may help to see my previous post for background information on Bell's Theorem. However, I hope that this is understandable, even if you didn't read the background.
Quantum mechanics is well known for saying that particles can be in many locations at once, smeared out in a probability wave. Why can't we simply say that particles have definite locations, and that only our knowledge is complete? Because as Bell's Theorem shows, we will run into problems.
The set-up is that we have a source which emits pairs of entangled electrons. Each electron will go into a separate detector. These entangled electrons exhibit strange correlations in their behavior. If you measure the vertical spin of each electron, then both electrons are guaranteed to give the same result. However, we need not always measure vertical spin. We can also measure the horizontal spin, or any angle inbetween the horizontal and vertical. Let's tinker around with it.
We're going to use three different settings for the two detectors. Setting A will ask the question, "Is its spin going up?" Setting B will ask, "Is its spin going to the right?" Setting C will ask, "Is its spin going at a 45 degree angle between 'up' and 'right'?" If both detectors are on setting A, then they will both give the same answer for any given entangled pair of electrons. Likewise, they will give the same answer if they are both on setting B, or both on setting C.
Let us assume that each electron has predetermined answers to each of these three questions. We just don't know the answers yet! We can describe each electron as a set of three answers to the questions posed by detectors A, B, and C. For example, we could use (Y,Y,N) to denote an electron which will answer "yes" to A and B, but "no" to C. Similarly, (Y,N,Y) would denote an electron which will answer "no" to B, but "yes" to A and C.
Recall that if both detectors are on the same setting, then both electrons will give the same answer. This means that both electrons in a pair can be described with the same set of three answers. Each electron pair can be placed somewhere in this Venn diagram:
There are eight possible locations on the Venn diagram, but there's no reason to assume that each possibility is equally likely. Perhaps we could determine the probability of each possibility? But it's rather tricky. The problem is that we can only determine two of the answers for any given pair of electrons. Once each electron has been measured, we've changed their original quantum state and can no longer investigate it. Nevertheless, we can be mathematical detectives and figure it all out, right?
So let's say that we set the first detector to setting A and the second one to setting C. How often is the answer to A different from the answer to C? That is, how often do the electron pairs fall into the shaded region shown in figure (a)? The answer is about 15% of the time, or (1-cos(45°))/2. This can be predicted by quantum theory, but just as importantly, it can be directly observed with experiments.
What happens if we set the first detector to B, and the second detector to C, and we ask how often the two detectors give different answers? That is, how often do the electron pairs fall into the shaded region in figure (b)? Again, the answer is 15%.
How often do electrons fall into the shaded region in figure (c)? Figure (c) is the "union" of the two sets (a) and (b). All electrons which fall into the shaded regions in (a) and/or (b) will also fall into the shaded region of (c). This implies that electrons will fall into shaded region (c) no more than 30% of the time.
(a) Detectors A and C give different answers: 15%
(b) Detectors B and C give different answers: 15%
(c) The union of (a) and (b): At most 30%
Now, what happens if we set the first detector to A and the second detector to B? How often will the two detectors give different answers? If they give different answers, then the electron pair must fall somewhere in the shaded region of (c). Therefore, it couldn't happen more than 30%, right? Wrong! Both quantum theory and experiment show that the answers to A and B are different 50% of the time. We've run into a mathematical contradiction!
What we usually do when we find a mathematical contradiction, is we back up. We look around for assumptions. One of our assumptions must have been wrong! At the top, I said, "Let us assume that each electron has predetermined answers to each of these three questions." Under the mainstream interpretations of quantum mechanics, this assumption is wrong. Electrons do not have predetermined answers to every possible measurement. Instead, we must describe their states with probabilities.
I should also mention that there are ways around this conclusion. Perhaps I made other implicit assumptions, and maybe those assumptions are the wrong ones. There are several other ones, which have names like "realism", "locality", etc. But I'm not going to go into that. I'll just leave Bell's theorem as is, a proof that something is going intuitively wrong.