You will need to backup the database and files and transfer them.
Here's a basic outline:
1) Download all the files through FTP.
2) Backup the database (if it's really big it may need to be done through SSH, if not you can get an SQL dump from something like phpMyAdmin or the phpBB feature. Non-shell options are unreliable for large databases).
3) Upload the files on the new host.
4) Create a database and restore the data there (through SSH or your DB Admin tool).
5) Edit config.php to reflect the details of your new database (password etc).