While completely moving WordPress installation (including the database, files and plugins), widgets settings automatically moves and the website automatically starts working like before. This procedure is good for moving large websites but sometimes you might need need to reuse only widget settings, if there are only one or two widgets then that can be moved manually. However, if there are many widgets, widget areas in multiple sidebars then a better, fats and more easy solution is needed. This tutorial explains you how you can import & export widget settings across WordPress installations.
Install and activate the Widget Settings Import/Export plugin. After activation visit your admin area Dashboard -> Tools -> Widget Settings Export and select the widgets you like exporting and then click on ‘Export Widget Settings’ Button. Now the plugin will export all active widgets in widget_data.json file which can now be imported and the widgets can be restored using import feature.
So now you will be needing to install and activate Widget Settings Import/Export plugin on that website as well and then use Dashboard -> Tools » Widget Settings Import and click the Select File button.
All widgets will now be automatically restored to their appropriate widget areas in your website.