When you go into the Settings > Pernalinks section of WordPress, you can select a beautifully formatted links. When you do, it will generate what is called a .htaccess file with the ModRewrite rules needed in order to make your site work. Your old Podcast FEED URL used permalinks, so at some point you turned them off. Most likely you either did this unintentionally or something happened to the .htaccess file in your web site. Either way, if you're not sure where to proceed, I recommend yo contact your web hosting provider or the person who installed WordPress for you.