How does Whatsapp use my mobile fingerprint in their app?

I just updated to 2.20.64 to check

WhatsApp does need fingerprint permission, under " Use Fingerprint Hardware" and Use Biometric Hardware"

You can check under permissions in Play Store - see "others"

Edit : If you are wondering why the app doesn't ask you explicitly for permission, the reason is simple - both the permissions quoted above are normal permissions for which user approval is not required. Also see Permissions Overview

If an app declares in its manifest that it needs a normal permission, the system automatically grants the app that permission at install time. The system doesn't prompt the user to grant normal permissions, and users cannot revoke these permissions

(Emphasis supplied)

See this Stack overflow Question - Android nougat USE_FINGERPRINT permissions, which states the same