The plugin comes with a custom field called 'Story Field' that allows the story entered via the story section on the view issue page to be exported into XML, CSV, Word and other formats using standard Jira functionality while on the browser/search for issues page.
You can add this custom field via the issue field configuration screen in the Jira administration area as you would add any other custom field. First navigate to the Custom Fields page in the administration area and click on Add Custom Field button in the upper right.
In the pop up window that opens click on the Advanced tab and then enter 'Story' in the filter field at the top to show two custom fields provided by the plugin, select the one named 'Story Field' and click Next.
Enter desired field name and description (see suggested value in the screenshot) and click the Create button.
In the next window that opens where you normally associate the newly created custom field with different UI screens do ensure to leave all rows unchecked as this field is intended for exporting functionality only and not to be shown in any of the UI screens.
Once back on the issue search page, click on the export button in the top right of the page to show export drop down menu.
Click on the desired format for example 'CSV(All fields)' to generate and download the export file.
The generated CSV file will have a column titled 'Custom field (Story Field)' containing the user story entered via the plugin on the view issue page.