It only recognizes page, not the objects belong to it.
The application should be an open application because there is TestComplete icon on right side of the object tree (iexplore). Do I need to set up something or install something to make it work?
Re: TestComplete does not recognize Silverlight page
What version of TestComplete are you using? Please update to TestComplete 7.51 (if you are using an earlier version).
Besides that, I suggest that you refer to the "Preparing Silverlight Applications for Testing" help topic that explains how to prepare a Silverlight application for testing. Also, you need to add the "MicrosoftSilverlight" and "ATL:*" items and enable them in the Open Applications | UI Automation section of your project's properties. Please read the "Testing Silverlight Applications with TestComplete" help topic explaining how to do this.