Anthem's closed alpha will run from December 8th-9th on PC, PlayStation 4 and Xbox One.
This is why the initial test is only open for two play sessions across two days; it ends on 9 December. BioWare's goal with this alpha is to test matchmaking and server load, which is probably why the test will only be available in time-limited sessions instead of being open for a full 48 hours.
To sign up, simply head to the EA Community Playtesting webpage, sign in with an EA account, choose a platform and when you'd like to play the alpha, and submit.
It also explains why the alpha is closed, which basically means you have to be chosen or signed up to take part.
If you're registered for a Community Playtest, you'll get an email with instructions to download the client for the Closed Alpha before it begins. Please note that the Anthem closed alpha is "strictly confidential" so you will not be permitted to share captured media via streaming, social media and so on. Then you also choose the best timings for you from the available play test sessions.
According to the Electronic Arts support page, the "Anthem" closed alpha will occur for a limited time only from December 8 to December 9. The company even has made a new sign-up page for the fans who want to participate.
If you have Origin Access (Basic or Premier), EA Access, or you pre-ordered Anthem, you'll still get VIP access to a pre-launch demo. Now, one important thing to keep in mind: signing up does not mean that you'll get guaranteed access to the Alpha testing.