Mysterious shrine are special sites in Wasteland 2. Each of these shrines is dedicated to one of the 11 Kickstarter contributors who pledged $10,000 or more to the project.
All but one shrine can be discovered with a high Perception. Their locations are also revealed by characters in the game.
There are six shrines total. Each shrine adds a variable amount of experience points to the party. See this map for reference.
There are five shrines total, each with the same functionality.