file_get_contents() how to fix error "Failed to open stream", "No such file"

The URL is missing the protocol information. PHP thinks it is a filesystem path and tries to access the file at the specified location. However, the location doesn't actually exist in your filesystem and an error is thrown.

You'll need to add http or https at the beginning of the URL you're trying to get the contents from:

$json = json_decode(file_get_contents('http://...'));

As for the following error:

Unable to find the wrapper - did you forget to enable it when you configured PHP?

Your Apache installation probably wasn't compiled with SSL support. You could manually try to install OpenSSL and use it, or use cURL. I personally prefer cURL over file_get_contents(). Here's a function you can use:

function curl_get_contents($url)
{
  $ch = curl_init($url);
  curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
  curl_setopt($ch, CURLOPT_FOLLOWLOCATION, 1);
  curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, 0);
  curl_setopt($ch, CURLOPT_SSL_VERIFYHOST, 0);
  $data = curl_exec($ch);
  curl_close($ch);
  return $data;
}

Usage:

$url = 'https://...';
$json = json_decode(curl_get_contents($url));

Why don't you use cURL ?

$yourkey="your api key";
$url="https://prod.api.pvp.net/api/lol/euw/v1.1/game/by-summoner/20986461/recent?api_key=$yourkey";
$curl = curl_init();
curl_setopt($curl, CURLOPT_URL, $url);
curl_setopt($curl, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($curl, CURLOPT_FOLLOWLOCATION, 1);
curl_setopt($curl, CURLOPT_SSL_VERIFYPEER, false);

$auth = curl_exec($curl);
if($auth)
{
$json = json_decode($auth); 
print_r($json);
}
}

You may try using this

<?php
$json = json_decode(file_get_contents('./prod.api.pvp.net/api/lol/euw/v1.1/game/by-summoner/20986461/recent?api_key=*key*'));
print_r($json);
?>

The "./" allows to search url from current directory. You may use

chdir($_SERVER["DOCUMENT_ROOT"]);

to change current working directory to root of your website if path is relative from root directory.