It was a hotly debated topic for several decades. Until 1964 we weren't even sure we could ever resolve it, then a fellow named John Bell published a testable theorem showing a divergence between quantum mechanics and the classical explanation you have above (http://en.wikipedia.org/wiki/Bell's_theorem). In 1972, the first successful experiment of this theorem was performed, and incredibly, it showed entanglement is in fact more complicated than can be explained with classical mechanics http://en.wikipedia.org/wiki/Bell_test_experiments. The experiments have been repeated and refined over the years and little by little, the once common theory of local variables is now almost completely dead.
This even works if the states are known! This is how quantum cryptography works. You put two particles into Bell state with each other, give one to someone else, and then decide later what message you want to send.
I recommend learning the math. It is pretty simple if you already have linear algebra.
s_e= a |+-> + b |-+>
s_n= a |+-> + b |-+> + c |--> + d |++>
a|+-> + b|-+> is a very special-case entanglement. Yes, you'll see that one mentioned in Wikipedia (and quantum cryptography and etc) because it's very simple.
The general form for the states of two photons is exactly as you have listed for s_n. For some coefficient values of a, b, c, d, the photons are "entangled", for some, they are not "entangled". How do you know which is which? It is whether you can factor the polynomial into two separate expressions of the form (x|+> + y|->). If you can do this, the photons are not entangled; if you cannot do it, they are entangled.
Since most sets of (a, b, c, d) represent unfactorable expressions, you would expect almost any expression chosen at random to represent an entangled pair. Clean un-entanglement is the rare exception.
This is true only for pure-state entanglement (which itself is a rare exception ;-).
Then we look at one, and it is decided. If someone a second later looks at the other one, a light year away, it will be the opposite state. That has also been decided. Us looking at something over here caused a decision about what state something is in to be made over there, a light-year away. How did us doing something here affect a decision a light-year away? Two seconds ago the state of that particle a light year away was undecided; the "universe" had not given it a value. Then we did something here, and that particle a light year away had its value fixed. It did not have that value the whole time while we were moving the particles apart, but now it does.
The problem of local realism then arises from this smaller state space and the collapse of the wave function during the measurement.
That's based off what I know, though, I may be wrong.
There is nothing 'mysterious' or 'strange' or 'bizarre' about entanglement. Lets look at the classical version, suppose there is a red billiard ball at the middle of a billiard table. I shoot a white ball, such that one ball ends up in the upper left pocket and the other ball in the upper right. Then the state space is white ball in the left pocket and red ball in the right pocket or the white ball in the right pocket and the red ball in the left pocket. Because of momentum conservation it is impossible to have both balls in the same pocket and billiard balls also do not spontaneously change their color. So the moment I do look at one ball, I know the color of the other. It is exactly the same thing with 'quantum entanglement,' some conservation law demands that certain outcomes are impossible, and therefore the possible states of the system are restricted compared to a 'arbitrary' configuration. ( In the billiard example, I put one ball in one pocket. And then roll a dice two determine where to put the second.) But on the other hand, claiming that entanglement is somehow 'bizarre' lends itself to a Einstein quote. And we know that Einstein quotes are the pinnacle of modern science journalism.
[Edit] The parentheses are misleading. I mentioned a dice as a shorthand for arbitrary, not to imply that it has anything to do with some quantum process. So the point is, that in the classical case I follow a specific process, which leads to outcomes which are restricted compared to the case where I just put the billiard balls into arbitrary pockets.