You might check and see if the sound is muted on the video itself. I very often have this problem - I will log onto BGG, click on a video, and it will start playing with the sound within the video turned off, when the rest of my sound is fine, and I know I haven't muted anything recently. In the bottom left of the Youtube frame is a little speaker - if it shows an 'x', then you could be having the same problem.
Thanks for pointing that out. I am using a Kindle Fire tablet and I will check on the mute function.