The example they are giving is for requesting their API using cURL in the command-line. You can try that too.
In PHP you can also use cURL. You could try something like this:
<?php
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL,"https://api.nutritionix.com/v1_1/search");
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS,
"appId=id&appKey=key&query=milk");
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$response = curl_exec ($ch);
curl_close ($ch);
print_r($response);
?>
You should google cURL and PHP and read up on that.