Well, just so if anyone else runs into this problem,
I found one way to resolve it by using pi_RTEcssText()
function inside my extension file:
$outputText=$this->pi_RTEcssText( $value['bodytext'] );
where $value['bodytext']
is the string I get from the database-query in my extension.
This function seems to process data and return the full HTML (links, paragraphs and other tags inculded).
Note: If you haven't already, it requires to include this file:
require_once(PATH_tslib.'class.tslib_pibase.php');
on the top of your extension file.
That's it basically.