The plugin looks for the presence of a yarpp-template file (there are 6 sample templates provided with the plugin), find it at this location:
"wp-content\plugins\yet-another-related-posts-plugin\yarpp-templates".
You need to copy atleast one template file in root folder of your current/active theme (and yes you can copy multiple templates and they all will appear in the template-file dropdown in the plugin options in the Settings). Copy template file(s) and refresh the YARPP options page and the template-file dropdown should now get enabled.
You can further extend or customize these templates from the "Appearance > Editor" menu. All the YARPP template files (that you copied into your current theme folder will appear in the Templates list on the "Edit Themes" page.
Let me know if you still face issues. I am just migrating my blog from WP.com to WP.org (dreamhost) and encountered a similar issue with this plugin :).
This information is also present in the bundled readme.txt file with this plugin under "How can I use the custom template feature?" question.
Hope this is helpful.