Humans actually do, but they do it by drinking the reindeer piss after the reindeer have eaten it.

I imagine it's either there's a chemical in the new fungi that specifically makes the rams high but not us or there is something in there that doesn't affect them but is toxic to us

Based on a cursory Google search with the terms "bighorn sheep"+"Psychoactive fungi" it would seem they are referring to rumors of a psychoactive lichen that have not been formally identified.

-secondly, you don't want to eat Amanita Muscaria aka fly agaric mushrooms unless you have thorough knowledge of what you're doing. It contains a hepatotoxic compound (hence the deer piss reference in someone else's comment) -and it should not be confused with the Psilocybin containing mushrooms AKA magic mushrooms. They do different things in the brain—The more you know 🌈 🌟