For example if you want to transmit a song through FM, does the receiver's dish size actually increase the range in which he can detect these signals? if it does then how?
A bigger dish is simply able to "catch" more of the power being sent through the air. A 100mm parabolic reflector could light a cigarette from sunlight. But it would take a 3m reflector to boil a liter of water.
As Tony Stewart cautioned, however, broadcast FM (down around 100MHz) is rather lower than most practical parabolic reflector dishes can operate at. Although there is a bit of reflection happening in many TV/FM antennas with those multiple rods in a long array, etc.
FM has a such a long wavelength that you would need an enormous satellite dish for it to effectively focus the FM signal.
Thus gain is lost from the ratio of the undersized dish to half wave.
Well it does not. The only thing that changes is the intensity of the received signal. The frequency range remains the same how large the dish be. The length of the antenna has effect on the receivable signal and the calculations for the same can be found in net easily.