"You need to have a specific Flash Player installed to view video content in Steam. Please follow these directions to get the right version for Steam. You will need this Flash player even if you already have the latest player for your default web browser".
So here's how to get Adobe Flash Player to work with Steam for Linux on 64bit.
1. Download Adobe Flash Player 32bit tar.gz archive from HERE (yes, you'll need the 32bit version even though you're running 64bit) and extract it in your home folder.
2. Create the "plugins" folder inside the ~/.local/share/Steam/ubuntu12_32/ directory using the following command:
mkdir -p ~/.local/share/Steam/ubuntu12_32/plugins
3. Copy the "libflashplayer.so" file (available in the folder extracted under step 1) to ~/.local/share/Steam/ubuntu12_32/plugins - if you've extracted the downloaded Flash Player 32bit archive in your home directory, simply use the following command to copy it to the right folder:
Now you can delete the "install_flash_player_11_linux.i386" folder from your home directory.
4. Restart Steam for Linux and Flash should now work on 64bit:
Note: the instructions above have been tested on Ubuntu 12.10 64bit.