If you keep seeing a popup asking for your Apple ID, whenever you start the app, please make sure you are logged in to your Drops account by using email, Facebook or Google.
Logging in to Drops should make the Apple ID popup to stop showing up.
You see this popup because we need to check the Apple Store receipt of your purchase on the device first. Without checking the receipt, we can’t verify your purchases. When you log in to your Drops account, we can store the necessary information from your receipt on our server, attached to your account. Then there is no need to check it on device, therefore no need to ask for the Apple ID again.