What exactly is (Horn) Antenna Gain
The gain is an indication of directionality.  IF you have an isotropic emitter that sends , say 1 W, into complete sphere you are emitting radiation over \$4\pi\$ steradians solid angle.  If you make the antennae more directional and emit over a smaller area then the energy is concentrated (per steradian) in a smaller solid angle.  This increase is a gain, gain in an active amplifier is a dimensionless number, and so is this.