How does an LC network enable a shoot through state for an inverter
How does the LC network prevent any damage of the switch during the
  short circuit stage?


I prefer this schematic for a Z source converter because it contains the all-important diode (red circle): -



That diode is important because without it, when T1 / T2 and/or T3 / T4 turn-on to create the shoot-through condition, energy stored in the capacitors would be lost instead of being "allowed" to produce double the supply voltage. Maybe it might be easier to see it if re-drawn: -







simulate this circuit &ndash; Schematic created using CircuitLab

In quiescent conditions node A will be at 100 volts and node B will be at 0 volts. Node C will also be at 100 volts (assuming an ideal diode)

When the switch closes (representing the shoot-through condition) the capacitors are placed in series and so node C suddenly rises to 200 volts and reverse biases the diode. This means that the applied input voltage to the inductors is twice as high. As shoot-through progresses the current ramps up as per a normal booster circuit but with potentially twice the input voltage.


  What is the use of connecting the capacitors in an "X" like fashion in
  between the inductors?


Hopefully, with the redrawn diagram you should be able to see that now.