The wp.config file in the root of an installation contains the database name, user, and password. You will need to adjust those values to match the backup database.
I would install a fresh copy of WP on your hosting site, after copying all files down to a new folder your local computer. Then, after the new WP install and basic setup, copy that new install down to your local computer (into another folder) for another backup. (Don't use those files unless you carefully inspect them for malware code.)
Then adjust the database name, user, password values in the new wp.config file to match the new database (you would have previously copied that old backup database to your host).
At that point, you should be able to log into the new install, with the backup database contents. Then start the long process of restoring content.
While you are doing that, consider adding backup to your site tasks. And keeping things updated. Lessons learned, and all that.