Backup and restore data to new site

That sounds like you just want to migrate your whole site. You can look for a tutorial on how to move a WordPress site to a new location. There are many out there. If, for some reason, you want to just move our plugin data over, then all of the menus and items are stored as custom post types, so you would have to migrate just the those post types (fdm-menu and fdm-menu-item) from your wp_posts table. Also, all of the options are stored in the food-and-drink-menu-settings array in the wp_options table, if you wanted to migrate the settings.