Blizzard said it will choose players who have active subscriptions to the regular game based on a variety of factors to help them test the beta. It could be based on your PC, your commitment to the game, or just your luck. If you get in, you’ll likely receive an email, but if not, you can check your launcher. And remember: as with any beta, your progress will not be saved for when the game launches later this year.
So committed, in fact, that modern WoW players are trying the beta and reporting what seem like bugs today but what were actually intended functionality 13 years ago. This became such a common occurrence that Blizzard publicly posted a list of known non-issues called the "WoW Classic 'Not a Bug' List." For example, hitboxes for the Tauren player race are much larger than those of other races. In a modern game, this would be seen as a serious balance issue (see: Apex Legends). But it's what vanilla WoW was like, so it has been faithfully reproduced.
Occasionally it's also been proposed to re-randomize all the gear. We didn't have best in slot lists back then. People weren't starting out as a fresh level 1 character with a complete list of which dungeons to run at 60 in order to gear as quickly and efficiently as possible. Now, all that data is available. If gear were re-randomized then all of those lists would become useless and thing would be close to how they were. Which is a more important part of the "vanilla experience?" the fact that item X from boss Y has exactly Z stats on it, or the voyage of discovery? Personally I think that voyage is way more important. Do you remember how many points of which stat on the ring or whatever that Darkmaster Gandling drops? I don't. Would it kill your experience if the BiS for every class were in a different dungeon so people had to actually play to figure it out rather than simply consult a spreadsheet? I don't think it would. For that matter, it probably wouldn't be a bad idea to handle it intelligently so that the better gear is in later dungeons. I remember there because some weird cases where level 40ish gear was better than anything that dropped in level 60 dungeons.
To actually get access to the beta, which has already started and continue to add more players, you need to sign up via your Blizzard account management page. Under Games & Subscriptions, scroll down to Beta Access and visit the Beta Profile Settings page. Once there, you’ll see a grid of available betas you can opt into. Once you’ve checked WoW Classic, hit Update Preferences and you’re set.