>However, I don't understand how should I add code to the page that uses this control when the button is clicked.

The same way you'd hook any event handler:

// can add to Page_Load
 this.nameOfUserControl.Search_Clicked += this.NameOfMethodToHandleEvent;

// Some other method in the page.
private void NameOfMethodToHandleEvent(object sender, EventArgs e)
{
   // Code here
}