Why cast to an interface?

There is only one reason when you actually need a cast: When doc is of a base type of an actual object that implements IStorable. Let me explain:

public class DocBase
{
  public virtual void DoSomething()
  {

  }
}

public class Document : DocBase, IStorable
{
  public override void DoSomething()
  {
    // Some implementation
    base.DoSomething();
  }

  #region IStorable Members

  public void Store()
  {
    // Implement this one aswell..
    throw new NotImplementedException();
  }

  #endregion
}

public class Program
{
  static void Main()
  {
    DocBase doc = new Document();
    // Now you will need a cast to reach IStorable members
    IStorable storable = (IStorable)doc;
  }
}

public interface IStorable
{
  void Store();
}

If the object implements the interface explicitly (public void IStorable.StoreThis(...)) that casting is the easiest way to actually reach the interface members.

Because you want to restrict yourself to only methods provided by the interface. If you use the class, you run the risk of calling a method (inadvertently) that's not part of the interface.