how to use ajax to display from database and load more in wordpress plugin

Here is one approach, by using an iframe for the AJAX request it can be much easier to learn and debug your AJAX code… try this:

function videos_info() {
    global $wpdb; 

    // add LIMIT 3 here
    $videos = $wpdb->get_results( "SELECT * FROM wp_ifvideos ORDER BY id DESC LIMIT 3" , ARRAY_N);

    foreach ( $videos as $video ) {
        echo do_shortcode("");
        echo $video[1]. '<br>';
    }

    $ajaxurl = admin_url('admin-ajax.php');
    echo "<iframe style="display:none;" name="loadmorevids" id='loadmorevids' src="https://wordpress.stackexchange.com/questions/276825/javascript:void(0);"></iframe>";
    echo "<script>function loadmorevideos(offset) {
        document.getElementById('loadmorevids').src="".$ajaxurl."?action=video_ajax&offset="+offset;}</script>";    
    echo "<div id='morevideos-3'><a href="https://wordpress.stackexchange.com/questions/276825/javascript:void(0);" onclick='loadmorevideos(\"3\");'>Load More Videos</a></div>";
}

// For Logged in Users
add_action('wp_ajax_video_ajax', 'video_ajax');
// For Logged Out Users
add_action('wp_ajax_video_ajax', 'video_ajax');

function video_ajax() {

    global $wpdb;    
    $videos = $wpdb->get_results( "SELECT * FROM wp_ifvideos ORDER BY id DESC" , ARRAY_N);

    $offset = $_GET['offset']; $html="";
    foreach ( $videos as $i => $video ) {
        // limit to 3 videos using offset
        if ( ($i > ($offset-1)) && ($i < ($offset+2)) ) {
            $html .= do_shortcode("");
            $html .= $video[1]. '<br>';
        }
    }

    // append the new load more link
    $html .= '<div id="morevideos-'.($offset+3).'">';
    $html .= '<a href="https://wordpress.stackexchange.com/questions/276825/javascript:void(0);" onclick="loadmorevids('.($offset+3).');">Load More Videos</a></div>';

    // replace any single quotes with escaped ones
    $html = str_replace("'", "\\'", $html);
    // (or alternatively)
    // $html = str_replace("'", "&apos;", $html);

    // replace the previous loadmore link with the loaded videos
    echo "<script>parent.document.getElementById('morevideos-".$offset."').innerHTML = '".$html."';</script>";
    exit;

}