Hello Nikita Yaroshevsky,
Please, either remove the UpdatePanel control from your ItemTemplate or replace it with RadAjaxPanel control. As soon as I did the latter, the page opened as expected and the Rate server handler was executed as expected, when I clicked on the rating control. I have attached my test page for your reference. In case it does not help you, please open a new support ticket and send me a running test project that demonstrates the problem you have.
the Telerik team