Stateless authentication for Rest Api - php

Question

searching for a good authentication method for my rest api i came a cross this :
"What is stateless authentication?
Again, stateless means without state. But, how can we identify a user from a token without having any state on the server? Surprisingly, it’s very easy! just send all the data to the client.
So what would you store/send (send to client/network)? The most trivial example is an access token. Access tokens usually have a unique ID, an expiration date and the ID of the client that created it. To store this, you would just put this data into a JSON object, and encode it using base64."
Now, having a self-contained token, you will need to make sure that nobody can manipulate the data. For this you should sign it using MAC algorithm or any other digital signature method available.
This is a little confusing for me , how can i validate the access token when the request comes(nothing stored to match it), but i find it a good idea and i want to implement it, any advice will be very helpful.
My rest api is very simple I receive every request to index.php , then i create new object with the request class to analyze every element of the request.
request class looks like this :
<?php
class Request {
public $url_elements;
public $verb;
public $parameters;
public function __construct() {
$this->verb = $_SERVER['REQUEST_METHOD'];
$this->url_elements = explode('/', $_SERVER['PATH_INFO']);
$this->parseIncomingParams();
$this->format = 'json';
if(isset($this->parameters['format'])) {
$this->format = $this->parameters['format'];
}
return true;
}
public function parseIncomingParams() {
$parameters = array();
if (isset($_SERVER['QUERY_STRING'])) {
parse_str($_SERVER['QUERY_STRING'], $parameters);
}
$body = file_get_contents("php://input");
$content_type = false;
if(isset($_SERVER['CONTENT_TYPE'])) {
$content_type = $_SERVER['CONTENT_TYPE'];
}
switch($content_type) {
case "application/json":
$body_params = json_decode($body);
if($body_params) {
foreach($body_params as $param_name => $param_value) {
$parameters[$param_name] = $param_value;
}
}
$this->format = "json";
break;
case "application/x-www-form-urlencoded":
parse_str($body, $postvars);
foreach($postvars as $field => $value) {
$parameters[$field] = $value;
}
$this->format = "html";
break;
default:
break;
}
$this->parameters = $parameters;
}
}
?>
After this i proceed with the proper controller that is the first element after index.php/
Thank you very much for your time and sorry if the question is not very clear as i am new to rest :/