The ReflectInsight framework only receives messages from their extensions (i.e., in your case NLog extension). From your application, when it sends log messages via NLog, the RI framework will route them to its configured listeners configured in your application. If no listeners have been configured, it will default to the Live Viewer as shown in your screenshot.
You can however, configure your application to store RI messages to a binary file with rotations configured. This way, if any problems do occur in production, you can request the logs from your operations and filter the issues from there.
Our future product will have a web version (HTML5) to allow you to view RI messages stored specifically in the database. However, the database will be proprietary format specifically for RI.
If you have any further questions, please don't hesitate to ask or provide feedback here: http://reflectsoftware.uservoice.com/
DISCLAIMER: I'm one of the developers of ReflectInsight.