I'm not exactly sure of the cause, but the solution is to specify the configuration setting to point to the correct folder.... If you add this line into your html editor code:
$CKEditor_config["filebrowserBrowseUrl"] = "../webassist/kfm/index.php?uicolor=".urlencode(isset($CKEditor_config["uiColor"])?str_replace("#","#",$CKEditor_config["uiColor"]):"#eee")."&theme=webassist_v2";
it should start working properly. I've done it for you on that page.