OK, then let's back up a little. I'm not sure I know how to state things 'in plain English', but I'll give it a shot.
First, let's define some terms. A physical system S is for now assumed to be a sequence of physical states si, occurent at times ti. The system traverses or evolves through its states---this is the origin of any observable change. A measurement can be regarded as a 'question' asked of the system; there are certain such questions that the system will always answer with either 'yes' or 'no', i.e. for any given state s[sun]i, the answer to C can only be either +1 (yes) or -1 (no). We can call these states then yes-states or no-states.
The rest is essentially simple algebra, and no further assumptions will be introduced. All that operationally is done, is to measure the system at four different points in time, and note whether it answers 'yes' or 'no'. This will give us a list of four numbers, such as, for example
[math]
C_1 = +1\\ C_2=-1 \\ C_3 =-1\\ C_4=+1[/math]
From these four numbers, we calculate the quantity
[math]X=C_1C_2+C_2C_3+C_3C_4-C_1C_4[/math]
As you can see for yourself, in the given example, X = -2. For another example, consider
[math]
C_1 = +1\\ C_2=+1 \\ C_3= +1\\ C_4=+1[/math]
Here, X = 2. If you want to, you may play around with these numbers; you'll find that there is no case in which X exceeds 2. This is then the inequality
[math]\langle C_1C_2\rangle+\langle C_2C_3\rangle+\langle C_3C_4\rangle-\langle C_1C_4\rangle \leq 2[/math]
which in words just means '(the average of) X is always smaller or equal to 2'. (Here, we take the average for technical reasons that need not bother us; it's clear that if the inequality is valid in every individual case, it's valid on average, as well.)
But now consider what it would mean if X were to exceed 2. A possibility would be
[math]
C_1C_2 = +1\\ C_2C_3=+1 \\ C_3C_4= +1\\ C_1C_4=-1[/math]
where I've now tabulated the products of results, not the resulty themselves. As you can see, this amounts to X = 4. How could this come about?
Well, in order for each term to be +1, the factors must be equal either both +1 or both -1. In order for a term to be -1, they must be different, one +1, and the other -1. So the above table implies that
[math]C_1=C_2,\,\,C_2=C_3,\,\,C_3=C_4,\,\,C_1\neq C_4,\,\,[/math]
But this, as I've already argued, is impossible to satisfy under the assumption that there is one definite state at any given time: this assumption necessitates that C_1=C4 if the first three of the conditions are to be satisfied, in contradiction to C_1=/=C4, the fourth condition.
But how can this be? Well, one possibility is provided by quantum mechanics: that of linear superposition. There, you find that whenever there are two states |s_1> and |s_2> possible, there is also a state like |s_1> + |s_2>. Then, it becomes possible to violate the inequality; and this violation is what we actually observe.
This is contrary to everyday logic, certainly. But you can find logics with which it is perfectly consistent; Birkhoff and von Neumann have proposed a logic in which the distributive law is not a theorem that is modelled by quantum mechanical propositions, and Putnam has (somewhat (in-)famously) argued that this is the 'correct' logic to reason about the world. On this logic, for instance, the conclusions you draw would by and large simply not follow. The logic we use everyday is, on this view, merely a classical approximation to the logic the universe actually observes.
Now I have some troubles with that explanation, resting mostly on what's sometimes called 'the principle of tolerance' (by Carnap, I think?): that different logics, in a much as they are consistent, really say the same things, and if they appear to disagree, then you simply got your interpretations confused. But consequently, I also think that classical logic does not have an a priori privileged place.
Ultimately, it's really similar to how you can't imagine higher-dimensional geometries: though some people claim to be able to, I never could visualize four orthogonal directions in space, let alone five or more. But that doesn't make them impossible---it's just a function of my own limitations, but those don't impose limitations on nature.
