Its not clear that this is true.
Working with animals has been a little disconcerting over the past 50-60 years. In the distant past, I think most evolutionary anthropologists and their like bought into the idea that humans were completely uniquely intelligent and spiritual. But the more we try to define human sensibilities apart from other animals, the more we find its difficult to see animals as being completely incapable of human feelings and even thoughts.
I refer you to this question which describes the shift in intelligence theories over the years: Why have humans evolved conciousness?
The above reference is just to say that every definition of intelligence that is fairly broad has been overturned by animals (thumbs, socialization, tool use, self awareness, communication and language).
If you want to say 'I know we're special, but I just can't say how' you are in lots of good company... it's become pretty hard to lay down a definition that includes all human beings and excludes animals.
Lately it seems as if the definition of spiritual experience is similarly hard to define. What is your definition? The capacity to receive and reciprocate affection or love is almost never brought up - that seems to be half of the internet. Yet, that's a common component to many ideas of spirituality. It was found a while ago that primates have some form of self-awareness to identify themselves as distinct from other individuals. True this was only demonstrating that they know themselves in a mirror, but other experiments show it probably is deeper than just that.
I think the most confounding evidence is the work with animals who have been trained from infanthood to communicate to humans through sign language or symbols. The most astounding example I know if is Koko the gorilla. She had the vocabulary of a 5 year old and the capacity to joke around, reason, and adopted pets as children including a cat called All Ball. When All Ball who was killed by a car she mourned and remembered her adopted cat for years. This page outlines all that, though there is a book that describes Koko's life.
The human staff sought to discover what Koko thought about death. So they asked, “Where do gorillas go when they die?” Koko answered, “Comfortable / hole / goodbye.”
They asked, “When do gorillas die?” Koko answered, “Trouble / old.”
They asked, “How do gorillas feel when they die: happy, sad, afraid?” Koko answered, “Sleep.”
So this has been taken as evidence that gorillas have a concept of their own mortality or possibly even life after death.
So what is left? Not clear to me. Any thoughts?
Some regions of the brain and control of electrical activity in the brain have been associated with spiritual experiences or acts like prayer and meditation. The temporal lobe is exciting because magnetic stimulation of that region can create the sensation that there is a presence in the room with you. there appear to be many regions which are active in different instances though, which is something you might expect.
There is no evidence yet that animals do not experience similar effects in their brains, though primates have very much the same structures as we do. Its too bad Koko has died - she might have told us.