It seems my explanations were not clear enough, so I am adding here an update with some more details.
The RadGrid integration with the RadClientDataSource works with its built-in UI. This means that you can enable filtering in the grid (set its AllowSorting and AllowFilteringByColumn properties to true) and it will filter the records in the browser. There will be no second request. I am attaching a short video for this.
Your second option (perhaps more suitable than the first if you do not want to use the grid's UI) is to bind the grid programmatically, which means providing a data source to it with your own code. We have a demo for this (LINK) and I am attaching a short video that demonstrates the difference. In this demo the built-in filter events of the grid are used to fetch new data, but you can do this in the OnClientItemSelected event of the dropdown list.
Your third option is to use the Kendo Grid and the Kendo DataSource client-side widgets. They are designed to operate entirely on the client, and the Kendo DataSource exposes the .filter() method that you can use (LINK). You can see the basics of creating and binding a kendo grid like this in our demos (LINK). For your convenience, I prepared a basic example (LINK).