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.
Analogy: think back on riding the swings in kindergarten. Was it fun? Now, imagine going back to those swings exactly as they were, and sitting in them now, as an adult. They're too small. They don't fit. Your feet drag on the ground because they're so low to the ground. The bar over your head is low enough that you can reach out and touch it, and even at the highest the swing will go, it's only about chest high when you're standing up. It's exactly the same swing, but riding on it now is a very different experience, yes?
FYI: If you’re returning to WoW after an extended absence prior to patch 5.0.1, it’s possible that you’ll have more than the maximum pet limit and/or more than 3 of a kind. However, this will be the only time that you’re allowed to surpass these limits. You will have to pare down your collection, eliminating your duplicates, in order to add new pets.