How to show video from wp option value?

Your code:

$cat_data = get_option("category_$cat_id");

if ( isset($cat_data['extra1'] ) ){
    echo do_shortcode('[videojs youtube="'.$cat_meta['extra1'].'" width="699" height="269" preload="auto" loop="true" autoplay="false" controls="false" class="responsive-video"]');
}

There’s a number of things you’re not doing or could do better here:

• You never check if the option exists, if get_option doesn’t find anything and returns a value of false your code may fatal error and crash
• You’re relying on the videojs shortcode, but you can remove this dependency by using the oembed functionality that comes with WordPress. Did you know if you place a youtube URL in content on a line by itself, it’s auto-magically converted into video player?
• You never check that the URL is indeed a URL, e.g. if the URL was: "]<script>alert('hacked');</script>[sldfjknv attr=", there’s a chance this could result in unescaped insecure javascript running on the frontend
• If said plugin or shortcode isn’t present the functionality will fail with no fallback
• It doesn’t account for empty values or strings of spaces, ” ” is not a valid youtube URL, but it is a valid string, and will generate an invalid shortcode
• If $cat_id is replaced with a variable that isn’t a number it could have unexpected consequences, and is a possible route to sql injection

With this in mind, lets refactor:

$data = get_option( 'category_'.$cat_id );

// does the object exist?
if ( $data !== false ) {
    // empty checks if the value is set, and also checks if it has something in it, not just spaces
    if ( !empty( $data['extra1'] ) ) {
        $url = $data['extra1'];

        // URLs are URLs, lets escape this URL so that nothing nasty is passed along
        $url = esc_url( $url );

        // grab the embed html using oembed ( it has a second argument to specify width and height but it should be unnecessary )
        $embed = wp_oembed_get( $url );
        // set up a fallback
        if ( $embed != false ) {
            $embed = '<a href="'.$url.'">'.$url.'</a>';
        }
        echo $embed;
    }
}

The new version checks the values are what they need to be, escapes the values, and uses the well tested WordPress Core APIs rather than a 3rd party. Responsiveness should be taken care of by your theme/plugins CSS. it also falls back to a hyperlink if the embed fails for whatever reason.