LC Resonance circuit : output voltage
Why is the voltage higher at the output? What is a intuitive way of
  explaining why the output voltage is higher than the input voltage. Or
  is it one of those circuit where it only starts to make sense when you
  drive the transfer function.


Inductors and capacitors store energy over time, this means that the voltage or current can get higher than the initial voltage or current:


Source: libretext: 14.5: Oscillations in an LC Circuit 

This only happens at the resonant point formed by the capacitor and inductor, the circuit wants to oscillate at a certain frequency (the Q point), and will do so. The energy in the system is preserved but you can get higher voltage.

This property of inductors is how a DC to DC converter works, we charge inductors to create a magnetic field, then switch them off which momentarily creates a higher voltage.