This is largely taken from this MSDN article but window.external can be used to allow your WebBrowserControl to execute public methods of your client Windows Forms application.
For example in your form you may have a function such as:
public void HelloFromTheForm()
{
MessageBox.Show("Hi client, thanks for calling me!");
}
And in the html loaded into your WebBrowserControl you may have a button that looks like:
<button onclick="window.external.HelloFromTheForm()">
Say hi to the form
</button>
So in regards to your question of 'Is this used to call the server side functions?', your form isn't 'server side' but it does allow you to call the C#/VB.NET code of your form from an embedded webpage.
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