There is (currently unused code) in the extension that demonstrates how it would work:
$localURL = SpecialPage::getTitleFor( 'UrlShortener' )->getLocalURL( [ 'url' => $fullURL ] );
It appears to not be working anymore:
Note that this will not produce a url directly, it merely presets the form so that users don't have to copy things from the addressbar manually. This would be a nice way for gadgets and local communities to expose the new shortener e.g. from a sidebar link. I assume that's why it was added :) but it looks like it may've broke in in a refactor during the past year.