In the YouTube admin, apart from immediately after you upload a YouTube video, there's no way to batch-view your video links, or more importiantly your video ID's. This Bookmarket is designed to work from the YouTube admin video manager page. On click, it'll replace the video title with "Video ID: YourVideoID".
Add a bookmark to your toolbar, for the url, copy and paste the below code. Then when you're on the correct page, click it and magic will ensue.
javascript:(function(){if(window.jQuery===undefined||window.jQuery.fn.jquery<v){var v="2.1.0";var done=false;var script=document.createElement("script");script.src="https://ajax.googleapis.com/ajax/libs/jquery/"+v+"/jquery.min.js";document.getElementsByTagName("head")[0].appendChild(script);script.onload=script.onreadystatechange=function(){if(!done&&(!this.readyState||this.readyState=="loaded"||this.readyState=="complete")){done=true;swapTitle()}}}else{swapTitle()}swapTitle();function swapTitle(){var LinksArray=$('.vm-video-title-container').find('.vm-video-title-content.yt-uix-sessionlink');LinksArray.each(function(){var YTnumber=$(this).attr('href').replace('/watch?v=','');$(this).html('Video ID'+': '+'<span style=\"color:red; font-weight:bold\"\>'+YTnumber+'</span>')})()}})();