Eisntein had a huge problem with the implication that quantum mechanics was a 'non-local' theory, meaning that if two particles are prepared in an entangled state then a measurement made on one particle will cause the wave-function of the other particle to collapse simultaneously, even if the two particles are separated by some (potentially large) distance. This, according to some interpretations, would violate causality by propagating a signal faster than light - since in special relativity, something travelling faster than light is effectively travelling back in time. Eisntein's phrase was intended to dismiss this non-locality as hokum, hence 'spooky'.
Interestingly,
subsequent experiments have shown that the universe really is non-local. However, there is a constraint on this non-locality, specifically that
it is impossible to send any kind of information across such a channel. So faster-than-light communication is not allowed and causality is ultimately respected.