I made a post quite a while back on "Quantum Entanglement for Dummies" that tried to describe what it is. I emphasized the fact that this phenomenon is different than classical physics because of one every important characteristics of quantum mechanics, which is the superposition principle that is built into the quantum wavefunction. So to be able to understand why quantum entanglement exists and why it is so "spooky", one must first understand the superposition concept.
Don Lincoln has produced a video on quantum entanglement, and if you pay attention closely, he starts off with describing the superposition concept and how that made a quantum system not "predetermined" before a measurement. He also give a good overview on a Bell-type measurement that shows how experiments agree with QM description but not the hidden variables scenario.
A good video to start you off on understanding this phenomenon.