This option will allow your GPU (graphics processing unit) to process certain video data, instead of your CPU doing it.
This is a good thing if your GPU will do a better job than your CPU - it will take load off the CPU and hopefully make the application run smoother.
However if your GPU is old and/or not very good, it may not be wise to enable this option because your CPU could in fact process the data quicker. I'd try it and see if there is a performance difference to decide if you want it enabled or not.
From henryg on the Steam forums:
It applies to all h.264 video content on Windows - certain game trailers, Vimeo or YouTube videos in the overlay browser, broadcast watching, and any other use of streaming video in the client outside of the In-Home Streaming feature.
(Also, this in general is what "hardware decoding" in terms of graphics will refer to, so you can watch out for it elsewhere.)