Every new message is stored in your database, that's why it's consuming more and more space! And yes, it's using the space on your host. You can use "auto pruning" to automaticly delete old posts, and save space.
restoring the database is only used if you lose it for some reason like a server crash, or something. or if you change hosts, you can reinstall the board on the new host and restore the old database to the new one to get all your old members, posts, etc. back.
you have to create a backup of your database before you can restore it using that backup.
it is a good idea to backup your database on a regular basis. when you do this you will have a file to keep on your computer. if the need arises you can then use that file to restore from.
everytime someone posts a message, or signs up at your board, your database will increase by the size of that message or sign up info, usually just a few bytes each.