How is this URL being redirected?

You can add ?offers=/http://tracking.affiliate-link.com&aff_id=123 after your link and then check in the destination page (in this case the homepage) if is set $_GET['offers'], and in the case that it is, redirect to the content of the var, which is the url.

In the WordPress system I would create a page called “Offers” that would be in example.com/offers. You can create a page template attached to this page, and there check if the link sent by get parameters is what you want, and in case it is, redirect the user.

A simple page template would be something like this:

<?php  /*   Template Name: Offers   */ ?>
<?php if(isset($_GET['offers'])){
    wp_redirect( $_GET['offers'] );
    exit;
} ?>