Class View is intended to provide a hierarchical view of the "symbols" defined in your project. For website projects, it will only list the referenced assemblies and any classes defined in App_code.
So, no, you aren't missing a setting. You're missing the point of the Class View window ! ;-) Maybe you should be looking in the Solution Explorer instead.
BTW, if you're having trouble navigating through code files, you should consider better organizing your files (in folders, for instance).
Edit (after a Bounty has been placed on this question... drool! ) :
Okay, I have VS 2005 SP1. I tried the following cases:
a. Create New Website (VB or C#):
The Class View window is blank until you add an App_Code
folder. When added, the Class View window displays any classes added to the App_Code folder and the default Project references (System, System.Web, etc). It does not however display any Page code-behind classes.
b. Create New Project -> Web -> ASP.NET WebApplication:
The Class View window auto-populates with the Project references and the Page code-behind classes. This happens before I added any folder containing class files.
I am not sure if you are familiar with the ASP.NET Web Application Projects. If not, you need to have the Service Pack for VS 2005 installed. You can get it here. Else, you can also install the Web Application Projects extension directly.