How do I enable the Twig extension?
This should just work. Make sure to follow http://twig.sensiolabs.org/doc/installation.html#installing-the-c-extension and run the commands from within /path/to/docroot/vendor/twig/twig. Next, you should check again that extension=twig.so exists in your php.ini file and that Apache has been restarted/reloaded.

To confirm that the extension is being loaded, you can run php -c /path/to/php.ini -i | grep twig