How it works...

When the application loads with the control on the UI, the first thing that it does is to load its defined template and call the OnApplyTemplate() method. OnApplyTemplate() is a virtual method present inside the System.Windows.FrameworkElement class, which gets invoked when application code or internal processes call the System.Windows.FrameworkElement.ApplyTemplate().

As you can see in the OnApplyTemplate() method implementation, it finds out the template child named PART_Button using the GetTemplateChild method call, and registers its associated Click event:

public override void OnApplyTemplate() 
{ 
    base.OnApplyTemplate(); 
 
    if (GetTemplateChild("PART_Button") is Button searchButton) 
    { 
        searchButton.Click += OnSearchButtonClicked_Internal; 
    } 
} 

The Click event then invokes the custom event (SearchButtonClick), passing the SearchTerm as SearchEventArgs. Now, when you click on the button in the application UI, it fires the OnSearchButtonClicked_Internal...