This div is used to provide a shadow for the popup calendar of the RadDatePicker, you can see it in our online demos, too. It has been part of the picker for quite some time now.
Calendar / DatePicker/First look
If you do not insist on having a shadow, you could just hide the div, as you did.
However, as the described behavior is not how the control usually behaves, I assume that there are some custom styles on your page which interfere with the calendar's display. If you would like to debug this behavior, you could try removing your styles to see if this fixes the issue and then start re-adding them one by one until you see which breaks the layout.
Alternatively, you could use a tool like Firebug
to track the styles applied to the RadDatePicker control and see which one exactly causes the problematic look.
Also, if you provide us with a live url and your markup and custom styles, we will inspect the problem, too.
the Telerik team
If you want to get updates on new releases, tips and tricks and sneak peeks at our product labs directly from the developers working on the RadControls for ASP.NET AJAX, subscribe to their blog feed