This seems to have happened after the latest Adobe Flash Player update. Might be an issue of compatibility with Firefox, since in Internet Explorer the problem doesn't appear. Haven't tried any other browsers yet.

To me, it seems that the two images that appear are similar to the ones used in making 3D movies, but instead of displaying them overlapped they are displayed separately. This is just a wild guess and I have no actual proof to back this up.

PS: I am also using Firefox 3.6.13 on Windows XP.

Later edit: I have found a workaround for this issue. Right-Click on any improper displayed video and choose Settings. On one of the tabs found there is a checkbox "Enable Hardware Acceleration". It was enabled for me, after disabling it and restarting the browser the video was back to normal.

Thank you adc_raptor! I followed what you did, and the videos work perfectly fine!