First, I am not a member of BGA staff, so some of this information may be out of date or otherwise inaccurate.
But...
Games are considered intellectual property, and as such have owners (generally the game publisher). Each owner has a different philosophy about online versions of their games. Some think that if people can play it online, they will not buy physical copies (and make money for the owner), and therefore do not allow any online versions. Some owners think an online version is OK, but only if under their complete control. Some other owners believe that online versions give them more exposure, and allow others to develop games. A rare few even actively seek online versions, and pay for the game to be developed.
Now, BGA generally works with the third type, those that aren't opposed to others developing the game. BGA must contact them and obtain the rights to start developing the game. Having random users contact the rights owners may compromise the ability to obtain rights, so please don't bug the owner.
Then, BGA needs to find a volunteer programmer to actually write the code. As far as I know, to date *every* game on BGA has been done by volunteers, as well as all of the site code as well. (OK, fine, a little bit ago one of the two site developers quit his other job and started drawing a paycheck from membership donations, but it's still a tiny bit of code written for pay compared to all the written code.)
After that, the rights owner can still say that they don't like the implementation, and call off the project. But, if not, then BGA can launch the game legally.
Of course, some games are in the public domain, and don't need rights, just coding. Chess, checkers, backgammon, etc., fall into this group.
So, long answer short (too late, I know), BGA does get legal rights from whoever owns the game, which is why they haven't been shut down with cease-and-desist orders yet. Side note, to those wondering "why isn't game XYZ which has granted the rights (or is in the public domain) not on BGA?" is that the code still needs to be written by volunteers.
