Why can I not access members of the base class without casting?

Visible is not a member of IHttpHandler, so you should not expect to be able to call it on such a variable. It's a member of Page, by way of Control, I believe.

When you cast a variable to one of the base types/interfaces of the object's class, you can only call members that are on the type of the variable (or base classes/interfaces of that type).

The IHttpHandler interface is just that - an interface. It has no knowledge of a Visible member since any class can implement this interface. What you ended up doing is casting to an object of a class that does inherit the Visible member.