Calm down
The app uses Apple receipt to validate purchase. Normally Apple won't require Apple ID during update (It is automatically downloaded with the update and first purchase normally). But this time it should be caused by some external reasons that Apple wants you to login to download the receipt (e.g., OS update, or App Store upgrade, or Apple ID switch).
Don't afraid, iOS is secure enough that no app can ever get your account information. The input dialog is outside the app. If you don't input your Apple ID, the receipt won't be downloaded. And the app will not function properly.