The way I understand it is, when you under the influence your seeing the world in a different- more powerful or spiritual sense. This needs to be done under the proper guidance, or you could hurt yourself.
My uncle told me this story, he is what you would consider a "shaman".
Basically he says that when he was young, he stole his relatives peyote (uncle I think) and him and his girlfriend ate all of it. They started to see the world in a different light. They could see the life force in the nature around them, they could see the "blood or energy" moving through the veins in the leaves. As it started getting more intense they started to loose their minds. After about a day they went to their uncle who sang them songs and prayed with them until they came down from their "high."
Also, we're not from one of the peyote using tribes, but some members have adopted the peyote culture in recent times. My uncle isn't part of this peyote culture, but this his interpretation (as a healer) of it.