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.
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.