Install lite.computer on your new Mac and paste in the same license key from your purchase email.
If you no longer use the old Mac, remove your license from it: open Settings, click Remove license in the Danger Zone, then Yes, remove my license. Then drag the app to the Trash. See Removing lite.computer.
Your pages live in the folder you chose, not inside the app. Move that folder, or keep it in iCloud Drive or Google Drive, and point the new install at it.
Settings the app kept on the old Mac, such as your photo and the Showspaces it listed, do not move with it.
Still stuck? Write to support@lite.computer.