This is a question I answered from a user on Quora. So many of our photography talk centers around technical things, I thought it a good change to focus for a moment on the more philosphical side of things...
This really is a profound question - perhaps even a book could be written about it! On one level, we can see that a photograph is a reflection of a moment in time. A moment that will never again be repeated in the entire existence of the Universe. When we look at that image, we are looking at something that no longer exists precisely in that form. It is as though we are traveling back in time to revisit that moment. Photographs in that way are like starlight - what we see now is that which has not existed for thousands, perhaps millions or billions of years - yet there it is for us to ponder, to contemplate, to absorb.
A photograph is also a reminder of the soul's presence - of the reality of a being that actually existed at that time and in some form still exists. A photograph can not happen without a photographer. In that way, a photograph is an expression of an immaterial mind, a being clothed in flesh as it were. It is a result of that being's interaction with the world around him or her.
On a larger scale, photography is merely one tool of creative human expression - the desire for all human beings to express themselves, to communicate their inner thoughts to the external world in an effort to say in essence "I am here!". And the mere act of saying "I am here!" shows the existence of that being as a separate, unique, self-aware entity.
But where does this creative expression come from? In its deepest sense, I believe that creative expression and self-awareness denote a profound truth - these things are an expression of the Divine. We ourselves are the creation of God and seek to express ourselves because we are made in God's image. We have in us the fingerprint of God. How is photography connected to man's soul? In the same way that our souls are connected to God.
In my faith, the Biblical book of Genesis says, "Then God said, 'Let us make mankind in our image, in our likeness, so that they may rule over the fish in the sea and the birds in the sky, over the livestock and all the wild animals, and over all the creatures that move along the ground.' So God created mankind in his own image, in the image of God he created them; male and female he created them.
As God observes, so we observe. As God thinks, so we think. As God creates, so we create, the greater to the lesser, a beautiful expression of our existence in this Universe and interaction with it.