HOWTO: Server-side access to Vimeo API with personal credentials by using official Vimeo's PHP library.

vimeo-get-video-data.php

// require the library
require '/path/to/vimeo.php/autoload.php';
use Vimeo\Vimeo;

function getVimeoVideoData( $videoId = '' ) {
  // set the authentication credentials
  $clientId = 'client_id'; // Client ID from application's dashboard
  $clientSecret = 'client_secret'; // Client secret key from application's dashboard
  $token = 'token'; // Personal token generated from application's dashboard
  
  // make the API call...
  $vimeo = new Vimeo( $clientId, $clientSecret, $token );
  $response = $vimeo->request( "/videos/$videoId" );
  
  // ...and check the response
  if ( isset( $response[ 'body' ][ 'error' ] ) ) {
    // the server has responded with an error code we should to handle
  } else {
    // now we have all the available informations about the required video
    // and we can process them as we wish.
  }
}