Activision has posted an FAQ for their upcoming Black Ops 3 Beta on their blog yesterday that reveals both the size of the game download, and how Beta tokens will be sent out.
The game file will be about 15GB, which is a fairly hefty file size for just a multiplayer beta. Regardless, be sure to make the necessary room on your preferred console or PC in preparation for the game.
For those that have already registered their Beta codes, Activision will be sending out Beta tokens starting next week. For PS4 players, your tokens will be arriving on August 19th, and for Xbox One and PC players, your tokens will show up on August 26th.
The actual FAQ is pretty extensive, so if you have any further questions, please check out the link above as Activision may have already answered it for you.