Use of apply_filter in plugin/widget class

I’m not sure if you are missing some code in the constructor where you write //code here :

function __construct() {
    //code here
 }

Try replacing it with:

function __construct() {
    parent::__construct(
       'byad_countdown_widget', // Base ID
        __('Byad Countdown', 'byad_countdown'), // Name
       array( 'description' => __( 'The Byad Countdown Widget', 'byad_countdown' ), ) // Args
    );
}

Your updated code example gives the following output:

<aside id="byadcd_widget-2" class="widget byad-countdown">
    <div class="countdown-display"  data-root="http://example.com/wp-content/plugins"></div>
    <h3 class="widget-title">you know what?</h3>
</aside>

on my 3.8.1 install with the TwentyTwelve theme.

So the filter seems to be working.