So, gain on a receiving antenna basically means the antenna can "hear" more in a certain direction, as opposed to being able to receive a signal equally at all points of a sphere around the antenna?
Gain on a receiving antenna shapes it's coverage pattern to cover more a certain direction?
I think that's a good analogy.
Imagine a megaphone--not the electric type, but the type that is just a cone with a hole in the end, like a cheerleader uses. It shapes the sound when you "transmit," but if you held it up to your ear, it would also shape the sound when you "receive." Antennas are the same, except since they work in the electromagnetic spectrum, they are much less intuitive to humans.
Great explanations - thanks to all who posted!