You can't yet.
Being "Greenlighted" in the movie industry (from whence the terms comes) and in Steam both mean that they've been accepted for further development. It doesn't mean that the game/movie is actually released and available for download / for watching in theatre, just that production with that company is moving forward and it very likely will be released at some point.
The Steam Greenlight page for Black Mesa shows that the game has been Greenlighted, meaning that it has received enough support from Steam users in the form of votes to go into production as a Steam-powered download. At the moment, there are only 10 Greenlighted games, among thousands of proposals on Steam Greenlight. (Zero games have been released through the Greenlight process yet (as of Sept 2012), which isn't surprising since the whole idea is new.)
How to get the game
I downloaded and started playing Black Mesa only two days ago, so the process is fresh in my mind.
To play Black Mesa, you need an installation of the Source SDK Base 2007 on Steam, which is free. To verify if you already have it, and to install it if you don't have it yet:
- Go to your Game Library in the Steam Client.
- Click on the "All Games" drop-menu next to the Search Bar.
- Pick "Tools" from the drop-down menu.
- Find Source SDK Base 2007 and verify if it has been installed. Install it, if it's not yet installed.
Then, download the blackmesa.zip file from the official homepage. I went with a Bittorrent download (because I guessed that my connection could handle more than I was likely to get from the site's direct-download link), and it downloaded in a bit under two hours (at an average of 1Mb/s).
Inside the zip is a
.7z archive and a
.exe installer. The installer found my Steam account, had it download and install the Source SDK, and then unpacked the files in the
.7z archive into the location in the Steamapps folder where Steam looks for Source mods. It handled this all automatically, which I must say is pretty slick for a mod. It then asked me if I wanted to launch Black Mesa right away, and notified me that it would show up in my Steam Library after I shut down and restarted Steam.
All in all, it was a painless install. It will be even easier when it moves out of Greenlight and becomes a Steam Release game, but it can't get too much easier than it already is.