It is 'spooky' correlation at a distance, not action. An analogy: Say you have two balls, red and blue. You have someone put them into black boxes (hidden from you) and give one to you and one to someone else. That other person takes theirs home. When you open your box, and 'take the measurement' and observe you have the blue ball, you 'instantaneously' know that the other person has the red one.
The quantum complication is that while the red and blue balls are in a definite state the entire time due to the infeasibility of isolating a macro-sized object like a ball from all other atoms and photons, the entangled particles are not in a definite state, depending on your interpretation of Quantum Mechanics.
There are 3 major interpretations of quantum mechanics, and this has different implications for each. In the Copenhagen interpretation, the measurement causes the quantum wave function to 'collapse' into a particular state. This is generally where the term spooky comes from, since that came out of the Bohr/Einstein discussions. In the non-local hidden variables interpretation, then some hidden variable that acts non-locally causes actual definite particles to travel through the quantum wave probabilities (ie. the particles where never in an indeterminant state). In the Many Worlds interpretation, the universe enters into a superposition of both basis states (|you observing red, ball is red> and |you observing blue, ball is blue>, you are simply part of the quantum interactions), 'branching' so to speak, but that doesn't affect the fact that in each branch observing red means you know the other is blue and vice versa.
Someone else mentioned Bell's Theorem. Bell's theorem only rules out local hidden variables, not non-local hidden variables. since it is one of the major interpretations in quantum mechanics that is still viable, though those non-local effects can be considered spooky.
Just finished The Teaching Company's QM video series, so blame them :P