I have never found a good, simple analogy. The best I've got is still a mathematical explanation, but I think it can make it "click" for some people: You have

I have never found a good, simple analogy. The best I've got is still a mathematical explanation, but I think it can make it "click" for some people: You have two photon detectors that you can rotate; one is in New York and one is in Paris. Two photons are entangled and one is sent to New York for measurement, and one is sent to Paris for measurement. What you find is that while each photon's polarization measurement is random, if you take the "random" values you get for both and compute their correlation over many trials, this value is a function of the difference in angles between the photon detector in New York and the photon detector in Paris. But here's the kicker: each photon detector can be rotated while the photons are in flight, before they get there. So how is it that the correlation is a function of both photon detector angles then? Especially considering it is light that you're measuring, and nothing can travel faster than light. That's why it's called spooky action at a distance. (Edit: Let me know if this makes any sense. I'm in a continual search for the simplest explanation of this phenomenon.)