How to query posts by category with the_title();

Look at the declaration of the function the_title(), it prints the title, if the third parameter $echo is set to TRUE which is the default:

/**
 * Display or retrieve the current post title with optional content.
 *
 * @since 0.71
 *
 * @param string $before Optional. Content to prepend to the title.
 * @param string $after Optional. Content to append to the title.
 * @param bool $echo Optional, default to true.Whether to display or return.
 * @return null|string Null on no title. String if $echo parameter is false.
 */
function the_title($before="", $after="", $echo = true) {
    $title = get_the_title();

    if ( strlen($title) == 0 )
        return;

    $title = $before . $title . $after;

    if ( $echo )
        echo $title;
    else
        return $title;
}

So either you use the_title( '', '', FALSE ) or even better get_the_title() because it makes the code better readable:

$title = get_the_title();
$args = array(
    'post_type' => 'Testimonials',
    'order' => 'ASC',
    'category_name' => $title
);