Speaking very generally, there are two ways to understand that entanglement experiment. One is that nothing changed when you measured and the states always had the value you measured, you just didn't know about it. This is known as hidden variables.

The second is that not only don't we know the state of the particles, they don't fundamentally have a determined state until you measure them.

The example with red and green balls given in this thread is clearly of the first type. In fact a very primitive type of hidden variables that could easily be proven wrong. The balls are not actually quantum (it's an analogy so that's fair). Now, this is obviously the most intuitive, so why don't everyone agree on this? The problem is at the very heart of all quantum weirdness. Take the double slit experiment. If the particle really had a determined choice of slit prior to going through the slit, why are we seeing an interference pattern?

Now, you can come up with creative explanations that still have hidden variables, but then you run into something called Bell's Theorem which states that if you have hidden variables, the universe must be able communicate faster than light(non-local). In the end, there are some people who believe hidden variables is the correct way of looking at it. Since the math is exactly the same as the variables are completely hidden for us, we have no way of determining who's right. At least for the time being.

Bell's theorem says we have to give up one of three things:

  • Hidden variables
  • Locality: Local action cannot influence a system far away faster than the speed of light
  • Free will: It makes sense to talk about what would have happened if you had chosen to do something else.

It's not correct to say that hidden variables have been proven false. It's almost correct so say that local hidden variables have been proven false as we don't usually discuss free will. To further look at this let's say I have a bag of hexagons. The top three sides are all black, and the bottom are all white.

  1. If I measure any random side. I will get white half the time and black half the time.
  2. If I measure two opposite sides, I will get two opposite colors.
  3. If I measure two sides next to eachother. I will get opposite colors one out of three times.

But, For a quantum hexagon:

  1. If I measure any random side. I will get white half the time and black half the time.
  2. If I measure opposite sides, I will always get opposite colors.
  3. If I measure two sides next to eachother. I will not get opposite colors one out of three times. It will be slightly less.

Suppose you have a system of two arrows, which have to point in opposite directions, and you're guaranteed to measure one of them pointing up.

Now, let's say you disturb the system a little bit. You put in some "paint", which will paint an upwards pointing arrow red and a downwards pointing arrow blue. If the states are predetermined, one arrow will get all the red and one arrow will get all the blue. If they aren't, both arrows will get some of both colors.

In quantum systems, for certain kinds of arrows and "paint", the second result happens.

That means that some of the sides don't have a color until you actually measure it, but opposite colors always have the same color.