Basically what they do is create a huge number of entangled particles, separate each pair into locations A and B, then measure each the state of all of the particles at both locations (this breaks the entanglement, but that's okay.)

The measurements at A and B appear perfectly random according to all the tests of randomness that we have. But when you bring the measurements from A and B together, you find that they are correlated -- each pair might be e.g. in the same state, or the opposite state, depending on how the entanglement was created. A and B can be arbitrarily far apart.

You might think, well that's easy to explain, when you created the entanglement it set the state of each at that point. But no, you can prove that isn't the case, and that it must be the case that the entangled particles both have an indefinite state until they're measured, and the measurement of one affects the state of the other across any distance. (The proof is called Bell's inequality, see this video for more: https://youtu.be/ZuvK-od647c)