How can AC current pass through the capacitor? There is no connection between charge plates and dielectric. Is it done due to breakdown? How is this possible?"

Alternating charged current arrives at one terminal on the cap. The load to ground is connected at the other terminal of the cap. Actually 'alternating' can be a confusing electrical term to some folks. Think of the charged current as a wave that delivers this positive current from the bottom to the top of the wave and retreats or falls off at the bottom of the other side of the wave. 
This condition is caused from the rotating action of the rotor turning inside the stator fields in a generator. The ground through the load is attracted to the positively charged electrons at the barrier of the cap at the top of the wave from the generator. 
Now, Think charge separation on the ground side. As the electrons become excited on the ground side of the cap at the barrier across from the positive charge (attraction), a vacancy of non-charged electrons has occurred downstream at ground. When the positive charge and current falls at the bottom of the wave from the generator side, the excited electrons at the barrier of the cap on the ground side are now positive (charge separation) and move to power the load and fill the vacancy to ground. This is also the reason for the delay of current and charge moving through a cap to ground. A larger barrier equals a longer delay . Nothing but magnetic field attraction between electrons has happened at the cap barrier. 
This is why a DC current cannot move through a cap. It has no wave. If we pulse the DC, we can get it to move through the cap, but at a lower efficiency than AC.