You need a video editing program that allows you to do this, dragging and dropping videos wherever you want. Personally, I use Premiere Pro, which lets you import the videos, and then place/size them wherever. So for a closing shot you would drag and drop the videos right beside each other and scale them to whatever size you want, add text and images, etc.
In order to make it a link to their other videos, they use the annotation feature provided by YouTube. This is something you do after uploading your video, the options to edit it show up before you finalize and post your video. I don't think you can link directly to external websites like Twitter and Instagram, but you can link directly to other YouTube videos and channels and such.
Also I'm sure there would be some people on the forum willing to put a closing title card like this together for you, just ask around.