the hash is generated randomly each time
No, the hash is always the same for a given input, salt value and iterations through which the hash algorithm is run (which is controlled by the cost parameter).
The concept I am having trouble understanding is how one would check the password input against the hashed value if it is stored in a database rather than the being known by the page at that point in time?
You would check the password input at login time, using the password provided by the user, and the salt and potentially number of times to apply the hash algorithm associated with that user. Once the password check is successful, use a session or other mechanism to keep the user logged in.
