You're right; this is nothing to do with spooky action at a distance.

But sympathetic response, or resonance, is how this device would work---just like a radio. Axions couple to electromagnetism; by listening in on different frequencies we can see if there is any "cosmic signal being broadcast" (meaning: axions suffusing the galaxy); the frequency of the signal corresponds to the mass of the axion.

In that sense, the search for resonances is a classic approach in particle physics; various particles were discovered by finding increased signals at particular collision energies. What's different here is that it isn't a resonance in a collision cross-section but an axion --> electromagnetism conversion.