A confirmation provides proof that something is true. An example might be when you receive confirmation that your flight will be on time, or you find out that something you thought might have happened actually did occur.
Confirmation is verification or final proof of something. In the church, confirmation is a rite in which a person if awarded full acceptance into — i.e., confirmed in — the religion, usually presented in a ceremony attended by family and friends. You can see how that ties in with the more general definition of substantiation of a fact or assertion. Confirmation is an obvious derivative of the verb confirm.