If you already purchased game time or a subscription, make sure you are logged in with the correct account email. If you want to install and play WoW Classic or Burning Crusade Classic, you are required to have game time or a subscription.
Modern WoW is free to download and play until level 20. Once you selected the correct folder, the Install button will become the Play button. If the app shows the Install button but you have already installed the game before, you don't have to reinstall. Click on Locate the game below the Install button to select the folder where you installed the game.
Modern World of Warcraft, WoW Classic, and Burning Crusade Classic are all installed through the desktop app.