To simply remove that, you can do a basic replace in /catalog/controller/common/seo_url.php
Find:
return $link;
Before it on a new line put:
$link = str_replace('index.php?route=common/home', '', $link);
Edit by TheBlackBenzKid: If you want full SEO just use this line instead of the above:
$link = str_replace('index.php?route=", "', $link);
Also make sure SEO URLs is turned on in the Admin panel of the store.