Your host will usually have a utility like PHPMyAdmin for accessing a database; this will work fine if you have a small forum (and the file will be under say 75mb on a fast connection); if not then you'll have to use say SSH to backup the database, which would be the most efficient way (and what my migration guide recommends you use).
There is no way to just "download" the database with this "fetch" program, it's stored separate from your files. You'll either need to hire someone to move it for you or learn (very fast) how to do it for yourself...
