Do I have to pull the hash out of the DB first and then compare it against what the user inputs?
Well, only if you want to check if the hash matches, I suppose.
Look, this is no different than MD5 hashing. You take a password, run it through a function, and get some kind of gibberish back. WordPress doesn't use MD5 anymore because it's been broken using rainbow tables and such.
$password = 'plaintext password';
$wp_hasher = new PasswordHash(8, TRUE);
That will output the hash that is supposed to be in the database.