How does the dish size increase reception range of radio signals?
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.