Question about Yubico YubiKey 5C Nano

No, the Youbikey is a touch sensor, not a fingerprint reader.
The functionality is needed for 2-factor authentication 2FA. For example, you can set up that for certain accounts (e.g. mail accounts that support 2FA, social media accounts or bank accounts etc.) you not only have to enter your username and password but also touch the key to log in. This ensures that even if your username or password combination is hacked, strangers cannot log in because they cannot touch the YubiKey on your computer after trying to log in from somewhere using a stolen username/password.