FM transmitter circuit working?
Your understanding is flawed but redeemable. Firstly there is a subtlety in this type of design that is usually missed by beginners. It's called the miller capacitance. That internal transistor capacitance occurs between base and collector and is actually alterable by the base-collector voltage. This means that an audio signal on the base (from the microphone) will alter this capacitance AND because this capacitance is massively connected to ground via C1, it also modifies the tank resonant frequency (via C3 or the battery's low impedance).

You might think about this and say but hey, if the base is held constant then this effect happens throughout each cycle of the oscillator's output voltage so what problems might this cause. Yes, it does and this basically causes a non-linear sinewave carrier to be produced but, it's still modulatable by the audio.


  This L1C1 is connected to a battery voltage - 9V. Thus, it will not
  have any losses


Sure there are losses but the capacitor C2 steals a bit of that energy and feeds it to the emitter to make it into an oscillator. Turning a simple LC into an oscillator means that the losses are replenished each cycle and so oscillation is sustained.


  Since Antenna is nothing but a wire with some resistance. But this
  wire isn't connected to anything. Thus, it is like an open circuit. If
  it is an open circuit then why does the signal passes to the antenna?


The antenna may at first seem to be just a wire but consider that at 100 MHz (typical FM broadcat band) the signal wavelength is about 3 metres and a quarter wavelength is about 75 cm. It's no coincidence that FM monopole antennas are about 75 cm long because they nicely "interface" with free space (air, a vacuum, the gap between earth and Voyager 1 and 2) and convert regular electricity nicely into an electromagnetic wave. 

And the theory could go on and on but I don't intend to do that because you need to look into this yourself if you are interested.