In order to export Invoices to Xero from Briefcase you should ensure that the client name used in Briefcase is setup as a contact in Xero and also that the name between Briefcase and Xero is the same.
You should also ensure that you have integrated Briefcase with Xero, for more information regarding this review the Xero Accounting integration article.
To batch export invoices from Briefcase to your accounting system you can go to the 'Invoices' tab. You can then use the filter menu to filter by 'Open' status invoices. You can also sort/group invoices to filter out the ones you do not want to export. Once you're ready, press the 'Click here to export * invoices" link.
If you would prefer to only export out individual invoices, you can do these by viewing the invoice you would like to export, press the action menu and selecting option 'Export Invoice'.