If an antenna is passive, where does antenna gain come from, and what is EIRP?

Gain is concentration, not amplification. The reference is an isotropic radiator, which spreads power equally over a full sphere. A real antenna sends more of the same power in some directions and less in others. A gain of 15 dBi, a factor of 31.6, means that in the direction of the main beam the power density is 31.6 times what the isotropic source would produce from the same input power. In other directions it is correspondingly lower, and the total radiated power is never more than what you fed in.

A torch reflector is the usual comparison: same bulb, brighter spot, darker everywhere else.

EIRP, equivalent isotropically radiated power, is the bookkeeping figure for that: transmitter power minus feed losses plus antenna gain, in dB terms. 20 dBm + 15 dBi = 35 dBm, about 3.2 W. It is the power an isotropic antenna would need in order to produce the same field strength in the peak direction. Nothing radiates 3.2 W; the figure is only valid on the beam axis.

Regulatory limits are often written as EIRP for exactly this reason, since it is what a receiver in the beam actually experiences.

Gain and beamwidth are tied together because the sphere has a fixed area. A full sphere is about 41,253 square degrees, so an ideal lossless antenna whose beam is θ1 by θ2 degrees wide has a directivity of roughly 41,253 / (θ1 × θ2). A 30° by 30° beam gives about 46, or 16.6 dBi; halving both beamwidths adds 6 dB. Real antennas come out somewhat lower because of sidelobes and losses.

The practical consequence is that a high-gain antenna has to be aimed accurately and mounted rigidly, while a low-gain one is forgiving.

The same figure applies on receive. Antennas are reciprocal: the pattern and gain for transmitting and receiving are identical at a given frequency. On receive, gain shows up as effective aperture, Ae = G × λ² / (4π), the area from which the antenna collects power from a passing wave. At 2.4 GHz (λ = 0.125 m) a 15 dBi antenna has an aperture of 31.6 × 0.0156 / 12.57, about 0.039 m².

That is why antenna gain at both ends counts in a link budget even though only one end is transmitting.

For reference points at the low end: a half-wave dipole has 2.15 dBi, with a doughnut-shaped pattern that is uniform around the wire and has nulls off the ends. Its length is a little under half a wavelength, roughly 143 / f(MHz) metres, so about 0.33 m at 433 MHz.

Gain figures also assume matching polarisation and a matched feed. Crossing a vertical antenna with a horizontal one can cost 20 dB or more, far more than the difference between two antenna models.