> How does it work this way? That is, how could it be the case that those measurements can be related, but you can't use that relation to transmit information?

In a simple example, both people measure for example the spin in the x axis of the particles that travel in the z direction. Each has a 50% probability of obtaining "up" and a 50% probability of obtaining "down". But in every case the get the opposite result, so each one has a random number generator that is synchronized with the other. But each one alone has only a random number generator.

Lest suppose that someone try to use that to create an intergalactic first person shooter for two players :). Using the "synchronized random number generator" it is possible to make the bots move exactly in the same way in both players computers, but they are only "synchronized random number generator" so it's impossible for one player to know what the other player has done (unless they wait until the information arrive in a conventional way, with a speed <=c, but that would be a very big laaaag).

But this is only part of the story, because this process can be simulated with a central "random number generator" that sends the signals to both players.

The strange property is that if both players "magically" decide begin to measure the spin in the y direction, they will have the same 50% chances and always get the opposite result. Another possibility that avoids magic is that one of the players continues measure spin in the x direction and the other players begins to measure the spin y direction. Now each one has a "random number generator" that is independent of the other "random number generator", so each player has no clue about what the other player result. It's not useful for a IFPS, but it's useful as a physic experiment.
(And one of the problems with quantum mechanics is that you can't measure the spin in both directions, you must choose one. It's a little more complicate, but I don't' want to enter into the technical details.)

But this is only part of the story, because this process can be simulated with a central "random number generator" that generate two random numbers and then sends the signals to both players, one for the x direction and one for the y direction. This is a simplification of the "hidden variable theory", that says that the particles "know" in advance what to do if they are measured in for the x direction and in the y direction, in spite of that you can't measure both.

In the experiments the idea is that the measurements/player decide which direction to use while the particles are flying, so they don't have enough time to communicate (at <= light speed) and agree what the result would be. They have to be in accordance from the start (hidden variable theory) OR they have to communicate faster than light OR something even more strange.

Really, to do the measurements it's possible to choose not only the x or y axe, but any direction in that plane. So for every direction each player has a "random number generator", but they are not independent. If one measure in the x direction and the other at 45º the probability that the results agree in some number in between 50% and 100%. The 50% is for orthogonal directions that have independent results, the 100% is for the same direction that has ever the same/opposite result, and for the other angles there is a formula to calculate the value. To simulate this you need a lot of hidden variables, or at least a few an a formula to calculate the result for each direction, or any other variation of this idea. There are many possible proposal, some are more simple and some are more complicated, so the idea is to put all of them in the "hidden variable theory" bag and forget for a moment the details.
The problem is that Bell proved that for any "hidden (local) variable theory" some inequality holds. This inequality ignores the details of the specific theory, so it's not possible to invent a more complex "hidden (local) variable theory" that breaks the inequality. When the same calculations are evaluated using the quantum mechanics the result is a value that for some angles is allowed by the Bell inequality but for other angles the value is forbidden by the Bell inequality. So there is a different prediction of some measurement using the quantum mechanics and any "hidden (local) variable theory".

And it is possible to do this experiment and calculate a value for every angle. And the result is that for the problematic angles the result agrees with the quantum mechanics predictions and don't holds the inequality that is predicted from any "hidden (local) variable theory". So we must eliminate all the "hidden (local) variable theory". (For the non problematic angles the result agrees with the quantum mechanics predictions again, and holds the inequality as expected because they are not problematic).

More details: http://en.wikipedia.org/wiki/Bells_theorem