In Opera 42.0.2393.94 i have the same problem, but this instruction solved the problem:
- Install the chromium-codecs-ffmpeg-extra library.
- Copy the libffmpeg.so file from the library (/usr/lib/chromium-browser)
- Replace the libffmpeg.so file in the browser folder (/usr/lib/x86_64-linux-gnu/opera) with this file.
- Restart the browser.
But in the new version (43.0.2442.806) it no longer works.
In this topic (http://forum.ubuntu.ru/index.php?topic=266804.0) written (forum in russian language): version of the file libffmpeg.so must be the same version of the engine chromium used browser Opera. In Opera 42.0.2393.94 chromium engine, I guess, 56.0.2924.76. But i can't find the chromium-codecs-ffmpeg the same version.
At page https://www.youtube.com/html5 seen that the problem is the H.264 codec: