Clean links in: the_author_meta(‘description’)

<?php
    // grab description,
    // note the "get_", we're not echoing the author meta, we're returning it
    $user_description = get_the_author_meta('description');
    // removing all HTML tags:
    echo strip_tags($user_description);
    // removing all tags apart from paragraphs:
    echo strip_tags($user_description,'<p>');
    // removing just anchors (i.e. <a> tags):
    echo preg_replace(array('{<a[^>]*>}','{</a>}'),array('',''),$user_description);
    // removing all links including their text (i.e. <a href="https://wordpress.stackexchange.com/questions/33568/...">...</a>):
    echo preg_replace('{<a[^>]*>(.*?)</a>}','',$user_description);
?>

Reference:

• strip_tags()
• preg_replace()