How does a wire loop know about magnetic flux change inside it, where there is no wire? I've always thought that a magnetic field must cross the wire to induce any change in it. But now I've learned that it actually doesn't have to even touch it, since the induced voltage in it is only dependent on the rate of change of magnetic field inside the loop of wire (doesn't matter if that field is actually changing inside the wires of the loop).
How is it that magnetic field that isn't even touching the wires induces a voltage inside of them? How does the wire "know" what the amplitude of magnetic field is in a space where there is no wire? This is beyond bizarre.
 A: The wire never 'knows' about anything outside of the wire itself. What really happens it that the changing magnetic flux induces an electric field locally, by
$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}.$$
Next, this changing electric field induces a magnetic field, by
$$\nabla \times \mathbf{B} = \frac{\partial \mathbf{E}}{\partial t}.$$
If the radius of the wire is $R$, then after time $R/c$, these induced fields propagate out to the wire, which feels the induced electric field. That's where the electromotive force comes from.

What I've just described is usually called 'light'. In situations where the wire is relatively small, you can ignore the light travel time and just use the equation
$$\mathcal{E} = - \frac{d \Phi_B}{dt}.$$
However, the underlying mechanism is still the light-speed propagation shown above. 
A: Well,
Nature wants things to stay the way they are.
Now if you change magnetic field (or magnetic flux) the nature will want to do something that does not let that change take place.
So if a magnetic field decreases in a loop, then an emf will show up and the current will start to flow in the wire in that direction in which the magnetic field of the wire adds up to the original magnetic field hence not letting it decrease. 
And if the magnetic field in a loop is increasing the current will flow in the wire in such a way that it doesn't let the increment take place.
As far as the direction of the induced emf is concerned, it points in the direction opposite to the change in the magnetic field in space.
