The library for phpWord can be found here: https://github.com/PHPOffice/PHPWord
An example for using phpWord Templates can be found here: http://phpword.readthedocs.org/en/latest/templates.html
To me it was obvious that it was not loading the template file because, following phpWords examples, if I add sections and text to the file, that section/text shows up in the output file. But nothing from the loaded template file shows up in the output file.
Error reporting is enabled to show all errors and warnings, but nothing is shown. While it is a bit annoying of syntax to me as well, you are actually suppose to save the $phpWord variable and not the created $document. Sections are added to the file in the same way, and saving them by referring to $phpWord does indeed work.
I have tended to stay away from the website you linked because it appears that the project has drifted from that website, as the version hosted there is very old compare to the current official version on github.