The reason for this is because Joomla's Search Engine Friend (SEF) URLS is turned on and the .htaccess file has not been modified to work with the temporary URL. To correct this simply:
Log in to the website via FTP ( see What are my FTP details? )
Download the file '/public_html/.htaccess' to your desktop
Open it and look for '# RewriteBase /'.
Change it to 'RewriteBase /~
Upload the file back to the '/public_html' folder
Remember to add a '#' infront of the 'RewriteBase' line when you switch to the live URL otherwise you will see a 404 error.
Alternatively you can switch off the SEF functionality within the Global Configuration while the website is being accessed via the temporary URL and switch on the SEF system when you move to the live URL.