I did a rather long post on the post "Quantum Entanglement for Dummies" which I tried to describe what it is. I emphasize the fact that this phenomenon is different from classical physics because of all the important features of quantum mechanics, it is the principle of superposition that is embedded in quantum wave functions. So, in order to understand why quantum fuss is and why it is so "scruffy", one must first understand the concept of superposition.
Don Lincoln presented a video on the quantum adventure, and if you pay close attention, he begins by describing the concept of superposition and how it happened that the quantum system was not "pre-defined" before measurement. He also gave a good overview of the Bell type measurement, showing how the experiments agree with the QM description, but not the scenario of the hidden variable.
It's a good video to get you started on understanding this phenomenon.