How AJAX calls work with TWIG

I am trying to understand how Twig can load a template through AJAX. From their website, it is clear how to load a template (http://twig.sensiolabs.org/doc/api.html)

echo $twig->render('index.html', array('the' => 'variables', 'go' => 'here'));

But how would this work for an AJAX call? How would you tell Twig that you want to 'render' something that is only a part of index.html ... and not reload the entire page? I looked at Twig's sole Ajax example (http://twig.sensiolabs.org/doc/recipes.html), but this doesn't explain how Twig knows what part of the page you want to change. Assuming your Ajax call results in page content updates. I just need a simple example of this, something more than what is on Twig's recipe page.